If `A` be any given square matrix of order n, then
`color{red}{A(adj A) = (adj A) A = | A | I}`
where `I` is the identity matrix of order n
`"Proof :"`
Let `A = [ (a_11 , a_12, a_13), ( a_21, a_22, a_23), ( a_31, a_32, a_33) ] ` , then adj `A = [ (A_11 , A_21, A_31), ( A_12, A_22, A_32), ( A_13, A_23, A_33) ]`
`=> "As we know Since sum of product of elements of a row (or a column)"`
`" with corresponding cofactors is equal to |A| and otherwise zero, we have"`
`A (adj A) = [ ( |A| , 0,0 ), ( 0, |A| , 0 ), ( 0,0, |A|) ] = | A | [ (1,0,0 ), ( 0,1,0 ), ( 0,0,1) ] = |A| I`
Similarly, we can show `color{orange}{(adj A) A = |A | I}`
Hence `color{orange}{A (adj A) = (adj A) A = | A | I`
If `A` be any given square matrix of order n, then
`color{red}{A(adj A) = (adj A) A = | A | I}`
where `I` is the identity matrix of order n
`"Proof :"`
Let `A = [ (a_11 , a_12, a_13), ( a_21, a_22, a_23), ( a_31, a_32, a_33) ] ` , then adj `A = [ (A_11 , A_21, A_31), ( A_12, A_22, A_32), ( A_13, A_23, A_33) ]`
`=> "As we know Since sum of product of elements of a row (or a column)"`
`" with corresponding cofactors is equal to |A| and otherwise zero, we have"`
`A (adj A) = [ ( |A| , 0,0 ), ( 0, |A| , 0 ), ( 0,0, |A|) ] = | A | [ (1,0,0 ), ( 0,1,0 ), ( 0,0,1) ] = |A| I`
Similarly, we can show `color{orange}{(adj A) A = |A | I}`
Hence `color{orange}{A (adj A) = (adj A) A = | A | I`