Name: Value of Determinant (Cofactor Expansion)
Uploaded: 2019-07-19
Duration: 2 min 56 s
Description: Multiply each element in any row or column of the matrix by its cofactor. The sum of these products gives the value of the determinant. Watch this video.

Question

Write Minors and Cofactors of the elements of determinant:

|\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}|

Accepted Answer

The given determinant is,

$|\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}|$

Minors of elements of a determinant are as follows:

$M_{11} = |\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}| = 1$

$M_{12} = |\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}| = 0$

$M_{13} = |\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}| = 0$

$M_{21} = |\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}| = 0$

$M_{22} = |\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}| = 1$

$M_{23} = |\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}| = 0$

$M_{31} = |\begin{matrix} 0 & 0 \\ 1 & 0 \end{matrix}| = 0$

$M_{32} = |\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}| = 0$

$M_{33} = |\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}| = 1$

We know that, cofactor of an element $a_{i j}$ , is denoted by $C_{i j}$ is defined by $C_{i j} = {(- 1)}^{i + j} M_{i j},$ therefore

Cofactors of elements of determinant are as follows:

$C_{11} = {(- 1)}^{1 + 1} M_{11} = 1 \times 1 = 1$

$C_{12} = {(- 1)}^{1 + 2} M_{12} = - 1 \times 0 = 0$

$C_{13} = {(- 1)}^{1 + 3} M_{13} = 1 \times 0 = 0$

$C_{21} = {(- 1)}^{2 + 1} M_{21} = - 1 \times 0 = 0$

$C_{22} = {(- 1)}^{2 + 2} M_{22} = 1 \times 1 = 1$

$C_{23} = {(- 1)}^{2 + 3} M_{23} = - 1 \times 0 = 0$

$C_{31} = {(- 1)}^{3 + 1} M_{31} = 1 \times 0 = 0$

$C_{32} = {(- 1)}^{3 + 2} M_{32} = - 1 \times 0 = 0$

$C_{33} = {(- 1)}^{3 + 3} M_{33} = 1 \times 1 = 1$

Write Minors and Cofactors of the elements of determinant:
$|\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}|$

Important Questions on Determinants

Learn and Prepare for any exam you want !

Write Minors and Cofactors of the elements of determinant: 100010001

Important Questions on Determinants

Learn and Prepare for any exam you want !

Write Minors and Cofactors of the elements of determinant:
$|\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}|$