Properties of Determinants: Definition, Properties, Examples

Properties of Determinants: A determinant is a particular number found using the square matrix. Properties of Determinants will help us simplify its evaluation by obtaining the maximum number of zeros in a row or a column. These properties are valid for determinants of any order. There are ten main properties of determinants, which includes reflection, all zero,  proportionality, switching, scalar multiple properties, sum, invariance, factor, triangle, and co-factor matrix property.

Determinants are used in a variety of ways in mathematics. For example, they are employed in shoelace calculations to calculate the area, which is advantageous because three collinear points define a triangle that is equal to 0. The Determinant is also utilised in multivariable calculi (most notably Jacobina) and in computing vector cross products. Continue reading this article to know more about the properties of determinants.

Determinant of a Matrix

Matrix refers to an ordered rectangular arrangement of numbers (in rows and columns) that are either real or complex. We denote the matrix, enclosing the elements by \([\,]\) or \((.)\)

A determinant is a particular number found using the square matrix. Properties of Determinants will help us simplify its evaluation by obtaining the maximum number of zeros in a row or a column.

The determinant of a matrix, say \(A,\) is denoted \(\det \,\left( A \right),\left| A \right|\) or \(\det \,A.\)

The determinant of a matrix is the scalar property of the given matrix. There are many applications of determinants. The determinant is used to find whether the matrix can be inverted or not. It is also used in the analysis and solution of simultaneous linear equations, also used in calculus, to find the area of triangles (if coordinates are given) and more.

Second-Order Determinant

A matrix with two rows and two columns are known as a second-order matrix. And its order is denoted by \(2 \times 2.\) The determinant of the second-order matrix contains the two terms, each of which is the product of two elements.
Example:
Let the square matrix of order two, \(A = \left[{\begin{array}{{c}} {{a_{11}}} \hfill & {{a_{12}}} \hfill \\ {{a_{21}}} \hfill & {{a_{22}}} \hfill \\ \end{array} } \right]\) then the determinant of the matrix \(A\) is calculated as follows:
\(\det \,A = \left|{\begin{array}{{c}} {{a_{11}}} \hfill & {{a_{12}}} \hfill \\ {{a_{21}}} \hfill & {{a_{22}}} \hfill \\ \end{array} } \right| = {a_{11}}{a_{22}} – {a_{12}}{a_{21}}\)

Practice Question From Properties of Determinants

Third-Order Determinant

A matrix with three rows and three columns are known as a third-order matrix. And, its order is given by \(3 \times 3.\)
The general method of finding the determinant of the \(3 \times 3\) matrix as follows:
1. First, consider the first-row element and multiply it by a secondary \(2 \times 2\) matrix, which comes from the elements remaining in the \(3 \times 3\) matrix that do not belong to the row or column to which your first selected element belongs.
2. Now, consider the second element of the first row and multiply it with the \(2 \times 2\) matrix, which excludes the second column and the first row entirely and accomplished by a negative sign.
3. Then, repeat the same for all the elements of the matrix, taking the sign alternatively.

Let the square matrix of the order three, \(A = \left[{\begin{array}{{c}} {{a_{11}}} \hfill & {{a_{12}}} \hfill & {{a_{13}}} \hfill \\ {{a_{21}}} \hfill & {{a_{22}}} \hfill & {{a_{23}}} \hfill \\ {{a_{31}}} \hfill & {{a_{32}}} \hfill & {{a_{33}}} \hfill \\ \end{array} } \right],\) then the determinant is given as follows:
\(\left| A \right| = \det \,A = \left|{\begin{array}{{c}} {{a_{11}}} \hfill & {{a_{12}}} \hfill & {{a_{13}}} \hfill \\ {{a_{21}}} \hfill & {{a_{22}}} \hfill & {{a_{23}}} \hfill \\ {{a_{31}}} \hfill & {{a_{32}}} \hfill &{{a_{33}}} \hfill \\ \end{array} } \right| ={a_{11}}\left|{\begin{array}{{c}} {{a_{22}}} \hfill & {{a_{23}}} \hfill \\ {{a_{32}}} \hfill & {{a_{33}}} \hfill \\ \end{array} } \right| – {a_{12}}\left|{\begin{array}{{c}} {{a_{21}}} \hfill & {{a_{23}}} \hfill \\ {{a_{31}}} \hfill & {{a_{33}}} \hfill \\ \end{array} } \right| + {a_{13}}\left|{\begin{array}{{c}} {{a_{21}}} \hfill & {{a_{22}}} \hfill \\ {{a_{31}}} \hfill & {{a_{32}}} \hfill \\ \end{array} }\right|\)

Shortcut Method

To calculate the determinant of the \(3 \times 3\) matrix, First, rewrite the matrix accompanied by a repetition of its two first columns, written outside the matrix at the right-hand side. The determinant value will result from the subtraction between the addition of products from all of the down-rightward multiplications of elements of diagonals and the down-leftward multiplications of elements of diagonals.

Properties of Determinants

There are ten main properties of determinants, which includes reflection, all zero,  proportionality, switching, scalar multiple properties, sum, invariance, factor, triangle, and co-factor matrix property.

Reflective Property of Determinants

The determinant remains unchanged if we interchange the rows to columns and columns to rows.

For a matrix:

\(\Delta = \left|{\begin{array}{{c}} {{a_1}} \hfill & {{a_2}} \hfill & {{a_3}} \hfill \\ {{b_1}} \hfill & {{b_2}} \hfill & {{b_3}} \hfill \\ {{c_1}} \hfill & {{c_2}} \hfill & {{c_3}} \hfill \\ \end{array} } \right| = \left|{\begin{array}{{c}} {{a_1}} \hfill & {{b_1}} \hfill & {{c_1}} \hfill \\ {{a_2}} \hfill & {{b_2}} \hfill & {{c_2}} \hfill \\ {{a_3}} \hfill & {{b_3}} \hfill & {{c_3}} \hfill \\ \end{array} } \right|\)

The determinant of the matrix and the determinant of the transpose of the matrix is the same.

\(\det \,A = \det \,{A^T}\)

Proportionality Property of Determinants

If any two rows or columns are identical (same) to each other, then the determinant of a matrix is zero.

Example:
The determinant of the matrix given below

\(\Delta = \left|{\begin{array}{{c}} a \hfill & b \hfill & c \hfill \\ a \hfill & b \hfill & c \hfill \\ p \hfill & q \hfill & r \hfill \\ \end{array} } \right| = a\left|{\begin{array}{{c}} b & c \\ q & r \\ \end{array} } \right| – b\left|{\begin{array}{{c}} a & c \\ p & r \\ \end{array} } \right| + c\left|{\begin{array}{{c}} a \hfill & b \hfill \\ p \hfill & q \hfill \\ \end{array} } \right| = {\mathbf{0}}\)

Switching Property of Determinants

If any two columns or rows are interchanged, then the sign of the determinant changes, keeping the absolute value of the determinant the same.

Example:

\(\Delta = \left|{\begin{array}{{c}} {{a_1}} \hfill & {{a_2}} \hfill & {{a_3}} \hfill \\ {{b_1}} \hfill & {{b_2}} \hfill & {{b_3}} \hfill \\ {{c_1}} \hfill & {{c_2}} \hfill & {{c_3}} \hfill \\ \end{array} } \right|\)
\( = {a_1}\left|{\begin{array}{{c}} {{b_2}} \hfill & {{b_3}} \hfill \\ {{c_2}} \hfill & {{c_3}} \hfill \\ \end{array} } \right| – {a_2}\left|{\begin{array}{{c}} {{b_1}} \hfill & {{b_3}} \hfill \\ {{c_1}} \hfill & {{c_3}} \hfill \\ \end{array} } \right| + {a_3}\left|{\begin{array}{{c}} {{b_1}} \hfill & {{b_2}} \hfill \\ {{c_1}} \hfill & {{c_2}} \hfill \\ \end{array} } \right|\)
\( = – \left\{{\begin{array}{{c}} {{a_1}} \hfill & {\left.{\left|{\begin{array}{{c}} {{c_2}} \hfill & {{c_3}} \hfill \\ {{b_2}} \hfill & {{b_3}} \hfill \\ \end{array} } \right| – {a_2}\left|{\begin{array}{{c}} {{c_{\mathbf{1}}}} \hfill & {{c_3}} \hfill \\ {{b_1}} \hfill & {{b_3}} \hfill \\ \end{array} }\right| + {a_3}\left|{\begin{array}{{c}} {{c_1}} \hfill & {{c_2}} \hfill \\ {{b_1}} \hfill & {{b_2}} \hfill \\ \end{array} } \right|} \right\}} \hfill \\ \end{array} } \right.\)
\( = – \left|{\begin{array}{{c}} {{a_1}} \hfill &{{a_2}} \hfill & {{a_3}} \hfill \\ {{c_1}} \hfill & {{c_2}} \hfill & {{c_3}} \hfill \\ {{b_1}} \hfill & {{b_2}} \hfill & {{b_3}} \hfill \\ \end{array} } \right|\)
\(\Delta = \left|{\begin{array}{{c}} {{a_1}} \hfill & {{a_2}} \hfill &{{a_3}} \hfill \\ {{b_1}} \hfill & {{b_2}} \hfill & {{b_3}} \hfill \\ {{c_1}} \hfill & {{c_2}} \hfill & {{c_3}} \hfill \\ \end{array} } \right| = – \left|{\begin{array}{{c}} {{a_1}} \hfill & {{a_2}} \hfill &{{a_3}} \hfill \\ {{c_1}} \hfill &{{c_2}} \hfill & {{c_3}} \hfill \\ {{b_1}} \hfill & {{b_2}} \hfill &{{b_3}} \hfill \\ \end{array} } \right|\)

Scalar Property of Determinants

If all the elements of any row or column are multiplied by any real number, then the value of the determinant of the matrix is also multiplied by the same real number.
Example:

\(\Delta = \left| {\begin{array}{{c}} {\lambda {a_1}} & {\lambda {a_2}} & {\lambda {a_3}} \\ {{b_1}} & {{b_2}} & {{b_3}} \\ {{c_1}} & {{c_2}} & {{c_3}} \\ \end{array} } \right| = \lambda \left| {\begin{array}{{c}} {{a_1}} \hfill & { {b_1}} \hfill & { {c_1}} \hfill \\ { {a_2}} \hfill & { {b_2}} \hfill & { {c_2}} \hfill \\ {{a_3}} \hfill & {{b_3}} \hfill & { {c_3}} \hfill \\ \end{array} } \right|\)
The determinate of the matrix \(\left|{\begin{array}{{c}} \lambda & {{a_1}} & {\lambda {a_2}} & {\lambda {a_3}} \\ \lambda & {{b_1}} & {\lambda {b_2}} & {\lambda {b_3}} \\ \lambda & {{c_1}} & {\lambda {c_2}} & {\lambda {c_3}} \\ \end{array} } \right|\) is \({\lambda ^3}\left|{\begin{array}{{c}} {{a_1}} \hfill &{{b_1}} \hfill & {{c_1}} \hfill \\ {{a_2}} \hfill & {{b_2}} \hfill & {{c_2}} \hfill \\ {{a_3}} \hfill & {{b_3}} \hfill & {{c_3}} \hfill \\ \end{array} } \right|\)
The property also called “trivial”.

Property of Sum

If all the elements of the determinant are expressed as the sum of two or more terms, then the determinant can be expressed as the sum of two or more determinants.

Example:

\(\Delta = \left|{\begin{array}{{c}} {{a_1} + {d_1}} & {{a_2} + {d_2}} & {{a_{3 + }}{d_3}} \\ {{b_1}} & {{b_2}} & {{b_3}} \\ {{c_1}} & {{c_2}} & {{c_3}} \\ \end{array} } \right| = \left|{\begin{array}{{c}} {{a_1}} \hfill & {{a_2}} \hfill & {{a_3}} \hfill \\ {{b_1}} \hfill & {{b_2}} \hfill & {{b_3}} \hfill \\ {{c_1}} \hfill & {{c_2}} \hfill & {{c_3}} \hfill \\ \end{array} } \right| + \left|{\begin{array}{{c}} {{d_1}} \hfill & {{d_2}} \hfill & {{d_3}} \hfill \\ {{b_1}} \hfill & {{b_2}} \hfill & {{b_3}} \hfill \\ {{c_1}} \hfill & {{c_2}} \hfill & {{c_3}} \hfill \\ \end{array} } \right|\)

Property of Invariance

Suppose any scalar multiples of corresponding elements of other rows or columns are added to every element of any row or column of a determinant. In this case, the value of the determinant remains the same.

The determinant will remain unchanged if we apply any operation of the form

\({R_i} \to {R_i} + k{R_j}\,or\,{C_i} \to {C_i} + k{C_j}\)

Example:

\(\Delta = \left|{\begin{array}{{c}} {{a_1}} \hfill & {{a_2}} \hfill & {{a_3}} \hfill \\ {{b_1}} \hfill & {{b_2}} \hfill & {{b_3}} \hfill \\ {{c_1}} \hfill & {{c_2}} \hfill & {{c_3}} \hfill \\ \end{array} } \right| = \left|{\begin{array}{{c}} {{a_1} + k{b_1}} & {{a_2} + k{b_2}} & {{a_3} + k{b_3}} \\ {{b_1}} & {{b_2}} & {{b_3}} \\ {{c_1}} & {{c_2}} & {{c_3}} \\ \end{array} }\right|\)

Zero Property of Determinants

If all the elements of any row or column are zeroes, then the value of the determinant equals zero.
Example:

\(\Delta = \left|{\begin{array}{{c}} 0 \hfill & 0 \hfill & 0 \hfill \\ a \hfill & b \hfill & c \hfill \\ p \hfill & q \hfill & r \hfill \\ \end{array} } \right| = 0\)

Singularity of Determinants

A matrix is said to be singular, whose value of the determinant equals zero.

\(\det \,A = 0\)

Property of Factors

By putting \(x = k,\) if the determinant becomes zero, then we can say that \(\left({x – k} \right)\) is the factor of the determinant \(\left( \Delta \right).\)

Property of Triangle

If all the elements below or above or both principal diagonal are zeros, then the determinant equals the product of principal diagonal elements.

\(\Delta = \left|{\begin{array}{{c}} {{a_1}} & {{a_2}} & {{a_3}} \\ 0 & {{b_2}} & {{b_3}} \\ 0 & 0 & {{c_3}} \\ \end{array} } \right| = \left|{\begin{array}{{c}} {{a_1}} \hfill & 0 \hfill & 0 \hfill \\ {{b_1}} \hfill & {{b_2}} \hfill & 0 \hfill \\ {{c_1}} \hfill & {{c_2}} \hfill & {{c_3}} \hfill \\ \end{array} } \right| = {a_1}{b_2}{c_3}\)

Property of Co-Factors

The determinant of any matrix is the same as the determinant of co-factors of all the elements.
Example:

\(\Delta = \left|{\begin{array}{{c}} {{a_{11}}} \hfill & {{a_{12}}} \hfill & {{a_{13}}} \hfill \\ {{a_{21}}} \hfill & {{a_{22}}} \hfill & {{a_{23}}} \hfill \\ {{a_{31}}} \hfill & {{a_{32}}} \hfill & {{a_{33}}} \hfill \\ \end{array} } \right| = \left|{\begin{array}{{c}} {{C_{11}}} \hfill & {{C_{12}}} \hfill & {{C_{13}}} \hfill \\ {{C_{21}}} \hfill & {{C_{22}}} \hfill & {{C_{23}}} \hfill \\ {{C_{31}}} \hfill & {{C_{32}}} \hfill & {{C_{33}}} \hfill \\ \end{array} } \right|\)
Where, \({C_{ij}} = \) co-factors of the element \({a_{ij}}\) in determinant.

Determinant of Identity Matrix

Determinant of identity matrix \(\left({{I_{n \times n}}} \right)\) of any order is one.
\(\det \,{I_n} = \left|{\begin{array}{{c}} 1 \hfill & 0 \hfill \\ 0 \hfill & 1 \hfill \\ \end{array} } \right| = \left|{\begin{array}{{c}} 1 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 1 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 1 \hfill \\ \end{array} } \right| = 1\)

Property of Multiplication

The determinants of multiplication or product of two matrices equal to the product of their individual determinants.

Let \(A\) and \(B\) are two matrices:
\(\det (AB) = \det A \times \det B\)

Property of Power

The determinant of a matrix with any power equals the determinant of the matrix raised to the same power.
\(\det {A^n} = {(\det A)^n}\)

Inverse Property of Determinants

The determinant of inverse of the matrix is the reciprocal of the determinant of the matrix.
\(det\left({{A^{ – 1}}}\right) = \frac{1}{{\det A}}\)

Solved Examples

Q.1. Using the Properties of determinants prove that \(\left| {\begin{array}{{c}} a \hfill & b \hfill & c \hfill \\ d \hfill & e \hfill & f \hfill \\ g \hfill & h \hfill & i \hfill \\ \end{array} } \right| = \left| {\begin{array}{{c}} b \hfill & h \hfill & e \hfill \\ a \hfill & g \hfill & d \hfill \\ c \hfill & i \hfill & f \hfill \\ \end{array} } \right|\)
Ans: Consider \(\left|{\begin{array}{{c}} a \hfill & b \hfill & c \hfill \\ d \hfill & e \hfill & f \hfill \\ g \hfill & h \hfill & i \hfill \\ \end{array} } \right|\)
Using the reflection property, interchange the rows and columns across the diagonal.
\( \Rightarrow \left|{\begin{array}{{c}} a \hfill & d \hfill & g \hfill \\ b \hfill & e \hfill & h \hfill \\ c \hfill & f \hfill & i \hfill \\ \end{array} } \right|\)
Using the switching property, interchanging columns \(2\) and \(3,\) then the sign of the determinant changes to negative.
\( \Rightarrow – 1\left| {\begin{array}{{c}} a \hfill & g \hfill & d \hfill \\ b \hfill & h \hfill & e \hfill \\ c \hfill & i \hfill & f \hfill \\ \end{array} } \right|\)
Using the switching property, interchanging rows \(2\) and \(1,\) then the sign of the determinant changes.
\(\Rightarrow ( – 1) \times ( – 1)\left|{\begin{array}{{c}} b \hfill & h \hfill & e \hfill \\ a \hfill & g \hfill & d \hfill \\ c \hfill & i \hfill & f \hfill \\ \end{array} } \right|\)
\( \Rightarrow \left| {\begin{array}{{c}} b \hfill & h \hfill & e \hfill \\ a \hfill & g \hfill & d \hfill \\ c \hfill & i \hfill & f \hfill \\ \end{array} } \right|\)
Hence, proved.

Q.2. Find the value of the determinant \(\left|{\begin{array}{{c}} 6 \hfill & 5 \hfill \\ 8 \hfill & 3 \hfill \\ \end{array} } \right|.\)
Ans: We know that determinant of \(\left|{\begin{array}{{c}} a \hfill & b \hfill \\ c \hfill & d \hfill \\ \end{array} } \right| = ab – cd,\)
So, the value of the determinant \( = \left|{\begin{array}{{c}} 6 \hfill & 5 \hfill \\ 8 \hfill & 3 \hfill \\ \end{array} } \right| = 6 \times 3 – 8 \times 5 = 18 – 40 = – 22.\)

Q.3. Show that \(\left|{\begin{array}{{c}} {{{\operatorname{cosec} }^2}\theta } & {{{\cot }^2}\theta } & 1 \\ {{{\cot }^2}\theta } & {{{\operatorname{cosec} }^2}\theta } & { – 1} \\ {42} & {40} & 2 \\ \end{array} } \right| = 0\)
Ans: Given: \(\left|{\begin{array}{{c}} {{{\operatorname{cosec} }^2}\theta } & {{{\cot }^2}\theta } & 1 \\ {{{\cot }^2}\theta } & {{{\operatorname{cosec} }^2}\theta } & { – 1} \\ {42} & {40} & 2 \\ \end{array} } \right|\)
By using the property of invariance, we can prove the above determinant equals zero.
Change all the elements of column \(1\) as by using: \({C_1} \to {C_1} – {C_2} – {C_3}.\)
\( \Rightarrow \left|{\begin{array}{{c}} {{{\operatorname{cosec} }^2}\theta – {{\cot }^2}\theta – 1} & {{{\cot }^2}\theta } & 1 \\ {{{\cot }^2}\theta – {{\operatorname{cosec} }^2}\theta – ( – 1)} & {{{\operatorname{cosec} }^2}\theta } & { – 1} \\ {42 – 40 – 2} & {40} & 2 \\ \end{array} } \right|\)
We know that \({\operatorname{cosec} ^2}\theta – {\cot ^2}\theta = 1,\)
\( \Rightarrow \left|{\begin{array}{{c}} {1 – 1} & {{{\cot }^2}\theta } & 1 \\ { – 1 + 1} & {{{\operatorname{cosec} }^2}\theta } & { – 1} \\ 0 & {40} & 2 \\ \end{array} } \right|\)
\(\Rightarrow \left| {\begin{array}{{c}} 0 & {{{\cot }^2}\theta } & 1 \\ 0 & {{{\operatorname{cosec} }^2}\theta } & { – 1} \\ 0 & {40} & 2 \\ \end{array} } \right|\)
If all the elements of any row or column are zeros, then the determinant equal to zero. Here, all the elements of column \(1\) are zeros.
So, \(\left|{\begin{array}{{c}} 0 & {{{\cot }^2}\theta } & 1 \\ 0 & {{{\operatorname{cosec} }^2}\theta } & { – 1} \\ 0 & {40} & 2 \\ \end{array} } \right| = 0\)
Hence, proved.

Q.4. Find the value \(x,\) of the singular matrix given as follows: \(\left[ {\begin{array}{{c}} {x – 1} & 4 \\ 1 & {x + 2} \\ \end{array} } \right].\)
Ans: Given \(\left[ {\begin{array}{{c}} {x – 1} & 4 \\ 1 & {x + 2} \\ \end{array} } \right]\) is a singular matrix.
We know that determinant of a singular matrix is zero.
\(\left| {\begin{array}{{c}} {x – 1} & 4 \\ 1 & {x + 2} \\ \end{array} } \right| = 0\)
\( \Rightarrow (x – 1)(x + 2) – 4 \times 1 = 0\)
\( \Rightarrow {x^2} – x + 2x – 2 – 4 = 0\)
\( \Rightarrow {x^2} + x – 6 = 0\)
\( \Rightarrow {x^2} + 3x – 2x – 6 = 0\)
\(\Rightarrow x(x + 3) – 2(x + 3) = 0\)
\( \Rightarrow (x + 3)(x – 2) = 0\)
\( \Rightarrow x = – 3\,or\,x = 2\)

Q.5. Show that, \(\det A = \det {A^T},\) for the matrix \(A = \left[{\begin{array}{{c}} 1 \hfill & 2 \hfill \\ 3 \hfill & 4 \hfill \\ \end{array} } \right].\)
Ans: Given: \(A = \left[{\begin{array}{{c}} 1 \hfill & 2 \hfill \\ 3 \hfill & 4 \hfill \\ \end{array} } \right]\)
Determinant of a given matrix, \(\det A = \left|{\begin{array}{{c}} 1 \hfill & 2 \hfill \\ 3 \hfill & 4 \hfill \\ \end{array} } \right|\)
\(\Rightarrow \det A = 4 \times 1 – 2 \times 3\)
\( \Rightarrow \det {\text{A}} = 4 – 6 = – 2\left( 1 \right)\)
We know that transpose of the matrix is obtained by interchanging rows to columns and columns to rows.
So, transpose matrix \({A^T} = \left[{\begin{array}{{c}} 1 \hfill & 3 \hfill \\ 2 \hfill & 4 \hfill \\ \end{array} } \right]\)
Determinant of transpose matrix, \(\det {A^{\text{T}}} = \left|{\begin{array}{{c}} 1 \hfill & 3 \hfill \\ 2 \hfill & 4 \hfill \\ \end{array} } \right|\)
\( \Rightarrow \det {A^{\text{T}}} = 4 \times 1 – 2 \times 3\)
\( \Rightarrow \det {A^{\text{T}}} = 4 – 6 = – 2\_\_\_\left( 2 \right)\)
From \((1),(2);\det A = \det {A^{\text{T}}}\)
Hence, proved.

Summary

In this article, we discussed the determinant of a matrix, which is the number obtained by solving the matrix. The determinant is valid only for square matrices of any order. Here, we studied the properties of determinants, which are reflexive property, all-zero property, property of sum, property of multiplication, scalar property, switching property, proportionality property, triangle property and property of invariance.

Frequently Asked Questions

We have provided some frequently asked questions about properties of determinants here:

Q.1. Why we use determinant and their properties?
Ans: There are many applications of determinants. The determinant is used to find whether the matrix can be inverted or not. It is also used in the analysis and solution of simultaneous linear equations, also used in calculus, to find the area of a triangle (if coordinates are given) and more.

Q.2. What is the determinant of a singular matrix?
Ans The determinant of the singular matrix is zero.

Q.3. How do you prove the properties of determinants?
Ans: The properties of the determinants are proved by using the square matrix, which is used for finding the determinants.

Q.4. State any three properties of determinant.
Ans: The three properties of the determinant are
\(\det A = \det \,{A^T}\)
\(\det {I_n} = \left|{\begin{array}{{c}} 1 \hfill & 0 \hfill \\ 0 \hfill & 1 \hfill \\ \end{array} } \right| = 1\)
\(\det {A^n} = {(\det A)^n}\)

Q.5. What is the relation between the determinant of a matrix and the determinant of the inverse matrix?
Ans: The determinant of inverse of the matrix is the reciprocal of the determinant of the matrix.
\(\det \left({{A^{ – 1}}} \right) = \frac{1}{{\det A}}\)

Q.6. Is \(\det AB = \det A\det B\)
Ans: Yes. According to the property of the multiplication of determinants, the determinant of the product of two matrices equals the product of their individual determinants.
\(\det AB = \det A \times \det B\)

You can also refer to the NCERT Solutions for Maths provided by academic experts at Embibe for your final or board exam preparation.

We hope this detailed article on the Properties of Determinants is helpful to you. If you have any queries on this article, ping us through the comment box below and we will get back to you as soon as possible.