Name: Properties of Determinants (Part 3)
Uploaded: 2019-07-19
Duration: 3 min 53 s
Description: Suppose we multiply a row of a determinant with a particular value, then what would be the change in the value of the determinant? Watch this video to learn...

Question

Find the adjoint of the matrix

A = [\begin{matrix} 1 & - 1 & 2 \\ 3 & 0 & - 2 \\ 1 & 0 & 3 \end{matrix}]

and also verify that

A (a d j A) = |A| I_{3} = (a d j A) A

.

Accepted Answer

Given matrix

$A = [\begin{matrix} 1 & - 1 & 2 \\ 3 & 0 & - 2 \\ 1 & 0 & 3 \end{matrix}]$

We have

$|A| = 1 (0 - 0) + 1 (9 + 2) + 2 (0 - 0)$

$= 0 + 11 + 0$

$= 11 \neq 0$

Co-factors of elements of coefficient matrix $A$ are as follows:

$C_{11} = |\begin{matrix} 0 & - 2 \\ 0 & 3 \end{matrix}| = 0 - 0 = 0$ , $C_{12} = - |\begin{matrix} 3 & - 2 \\ 1 & 3 \end{matrix}| = - (9 + 2) = - 11$ ,

$C_{13} = |\begin{matrix} 3 & 0 \\ 1 & 0 \end{matrix}| = 0 + 0 = 0$ , $A (a d j A) = |A| I_{3} = (a d j A) A$ ,

$C_{22} = |\begin{matrix} 1 & 2 \\ 1 & 3 \end{matrix}| = 3 - 2 = 1$ , $C_{23} = - |\begin{matrix} 1 & - 1 \\ 1 & 0 \end{matrix}| = - (0 + 1) = - 1$ ,

$C_{31} = |\begin{matrix} - 1 & 2 \\ 0 & - 2 \end{matrix}| = 2 - 0 = 2$ , $C_{32} = - |\begin{matrix} 1 & 2 \\ 3 & - 2 \end{matrix}| = - (- 2 - 6) = 8$ ,

$C_{33} = |\begin{matrix} 1 & - 1 \\ 3 & 0 \end{matrix}| = 0 + 3 = 3$

Therefore, the Co-factor matrix $C$ of the matrix $A$ is

$C = [\begin{matrix} 0 & - 11 & 0 \\ 3 & 1 & - 1 \\ 2 & 8 & 3 \end{matrix}]$

$Adj A = {[\begin{matrix} 0 & - 11 & 0 \\ 3 & 1 & - 1 \\ 2 & 8 & 3 \end{matrix}]}^{'} = [\begin{matrix} 0 & 3 & 2 \\ - 11 & 1 & 8 \\ 0 & - 1 & 3 \end{matrix}]$

Now $A (adjA) = [\begin{matrix} 1 & - 1 & 2 \\ 3 & 0 & - 2 \\ 1 & 0 & 3 \end{matrix}] [\begin{matrix} 0 & 3 & 2 \\ - 11 & 1 & 8 \\ 0 & - 1 & 3 \end{matrix}]$

$= [\begin{matrix} 0 + 11 + 0 & 3 - 1 - 2 & 2 - 8 + 6 \\ 0 + 0 + 0 & 9 + 0 + 2 & 6 + 0 - 6 \\ 0 + 0 + 0 & 3 + 0 - 3 & 2 + 0 + 9 \end{matrix}]$

$= [\begin{matrix} 11 & 0 & 0 \\ 0 & 11 & 0 \\ 0 & 0 & 11 \end{matrix}] = 11 [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] = |A| I_{3} . . . . . . . . . . . . . . (i)$

and $(adjA) A = [\begin{matrix} 0 & 3 & 2 \\ - 11 & 1 & 8 \\ 0 & - 1 & 3 \end{matrix}] [\begin{matrix} 1 & - 1 & 2 \\ 3 & 0 & - 2 \\ 1 & 0 & 3 \end{matrix}]$

$= [\begin{matrix} 0 + 9 + 2 & 0 + 0 + 0 & 0 - 6 + 6 \\ - 11 + 3 + 8 & 11 + 0 + 0 & - 22 - 2 + 24 \\ 0 - 3 + 3 & 0 + 0 + 0 & 0 + 2 + 9 \end{matrix}]$

$= [\begin{matrix} 11 & 0 & 0 \\ 0 & 11 & 0 \\ 0 & 0 & 11 \end{matrix}] = 11 [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] = |A| I_{3} . . . . . . . . . . . . . . (ii)$

From equation $(i)$ and $(ii)$ , It is proved that

$A (adj A) = |A| I_{3} = (adj A) A$ .

Find the adjoint of the matrix $A = [\begin{matrix} 1 & - 1 & 2 \\ 3 & 0 & - 2 \\ 1 & 0 & 3 \end{matrix}]$ and also verify that $A (a d j A) = |A| I_{3} = (a d j A) A$ .

Important Questions on Determinants

Learn and Prepare for any exam you want !

Find the adjoint of the matrix A=1-1230-2103 and also verify that A (adj A)=AI3=(adj A) A.

Important Questions on Determinants

Learn and Prepare for any exam you want !

Find the adjoint of the matrix $A = [\begin{matrix} 1 & - 1 & 2 \\ 3 & 0 & - 2 \\ 1 & 0 & 3 \end{matrix}]$ and also verify that $A (a d j A) = |A| I_{3} = (a d j A) A$ .