Name: Finding Inverse by Elementary Transformations
Uploaded: 2019-07-19
Duration: 12 min 5 s
Description: In this video, we will learn to find the inverse of matrices using elementary transformations of matrices. We will learn how to find the inverse of a matrix,...

Question

Find the matrix $X$ satisfying the equation

$[\begin{array}{l} 2 & 1 \\ 5 & 3 \end{array}] X [\begin{array}{l} 5 & 3 \\ 3 & 2 \end{array}] = [\begin{array}{l} 1 & 0 \\ 0 & 1 \end{array}]$

Accepted Answer

Let $A = [\begin{matrix} 2 & 1 \\ 5 & 3 \end{matrix}], B = [\begin{matrix} 2 & 1 \\ 5 & 3 \end{matrix}] and I = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$

$|A| = |\begin{matrix} 2 & 1 \\ 5 & 3 \end{matrix}| = 6 - 5 = 1 \neq 0$

Also $|B| = |\begin{matrix} 5 & 3 \\ 3 & 2 \end{matrix}| = 10 - 9 = 1 \neq 0$

Thus, $A$ and $B$ are invertible.

Cofactors of matrices $A & B$ are

$A_{11} = 3, A_{12} = - 5, A_{21} = - 1, A_{22} = 2$

$B_{11} = 2, B_{12} = - 3, B_{21} = - 3, B_{22} = 5$

Now $adj A = {[\begin{matrix} 3 & - 5 \\ - 1 & 2 \end{matrix}]}^{T} = [\begin{matrix} 3 & - 1 \\ - 5 & 2 \end{matrix}]$

And $adj B = {[\begin{matrix} 2 & - 3 \\ - 3 & 5 \end{matrix}]}^{T} = [\begin{matrix} 2 & - 3 \\ - 3 & 5 \end{matrix}]$

$A^{- 1} = \frac{1}{| A |} adj A = [\begin{matrix} 3 & - 1 \\ - 5 & 2 \end{matrix}]$

$B^{- 1} = \frac{1}{| B |} adj B = [\begin{matrix} 2 & - 3 \\ - 3 & 5 \end{matrix}]$

The given matrix equation becomes $A X B = I$

$A^{- 1} (A X B) B^{- 1} = I A^{- 1} B^{- 1}$

$(A^{- 1} A) X (B B^{- 1}) = A^{- 1} B^{- 1}$

$I X I = A^{- 1} B^{- 1}$

$X = A^{- 1} B^{- 1}$

$X = [\begin{matrix} 3 & - 1 \\ - 5 & 2 \end{matrix}] [\begin{matrix} 2 & - 3 \\ - 3 & 5 \end{matrix}]$

Therefore, $X = [\begin{matrix} 6 + 3 & - 9 - 5 \\ - 10 - 6 & 15 + 10 \end{matrix}] = [\begin{matrix} 9 & - 14 \\ - 16 & 25 \end{matrix}]$

Find the matrix $X$ satisfying the equation

$[\begin{array}{l} 2 & 1 \\ 5 & 3 \end{array}] X [\begin{array}{l} 5 & 3 \\ 3 & 2 \end{array}] = [\begin{array}{l} 1 & 0 \\ 0 & 1 \end{array}]$

Important Questions on Adjoint and Inverse of a Matrix

Learn and Prepare for any exam you want !

Find the matrix X satisfying the equation 2153X5332=1001

Important Questions on Adjoint and Inverse of a Matrix

Learn and Prepare for any exam you want !

Find the matrix $X$ satisfying the equation

$[\begin{array}{l} 2 & 1 \\ 5 & 3 \end{array}] X [\begin{array}{l} 5 & 3 \\ 3 & 2 \end{array}] = [\begin{array}{l} 1 & 0 \\ 0 & 1 \end{array}]$