Find (AB)-1 if A = \(A = \begin{bmatrix} {1} & {2} & 3 \\[0.3em] {1}

1.	Find (AB)-1 if A = \(A = \begin{bmatrix} {1} & {2} & 3 \\[0.3em] {1} & -2 & {-3} \end{bmatrix}\) \(M = \begin{bmatrix} 1 & -1 \\[0.3em] 1 & 2 \\[0.3em] 1 & -2 \end{bmatrix}\)
Answer» \(AB = \begin{bmatrix} {1} & {2} & 3 \\[0.3em] {1} & -2 & {-3} \end{bmatrix}\)\(\begin{bmatrix} 1 & -1 \\[0.3em] 1 & 2 \\[0.3em] 1 & -2 \end{bmatrix}\)= \(\begin{bmatrix} 6 & -3 \\[0.3em] -4 & 1 \\[0.3em] \end{bmatrix}\) Consider, \|AB\| = \(\begin{bmatrix} 6 & -3 \\[0.3em] -4 & 1 \\[0.3em] \end{bmatrix}\) =(6x1) - (-4 x -3) = 6 - 12 = - 6 ≠ 0 ∴ AB is a non-singular matrix ∴ (AB)^-1= \(\frac{1}{\|AB\|}\) (Adj AB) = \(\frac{1}{-6}\) \(\begin{bmatrix} 1 & 3 \\[0.3em] 4 & 6 \\[0.3em] \end{bmatrix}\)

Find (AB)-1 if A = \(A = \begin{bmatrix} {1} & {2} & 3 \\[0.3em] {1} & -2 & {-3} \end{bmatrix}\) \(M = \begin{bmatrix} 1 & -1 \\[0.3em] 1 & 2 \\[0.3em] 1 & -2 \end{bmatrix}\)

Answer»

\(AB = \begin{bmatrix} {1} & {2} & 3 \\[0.3em] {1} & -2 & {-3} \end{bmatrix}\)\(\begin{bmatrix} 1 & -1 \\[0.3em] 1 & 2 \\[0.3em] 1 & -2 \end{bmatrix}\)= \(\begin{bmatrix} 6 & -3 \\[0.3em] -4 & 1 \\[0.3em] \end{bmatrix}\)

Consider, |AB| = \(\begin{bmatrix} 6 & -3 \\[0.3em] -4 & 1 \\[0.3em] \end{bmatrix}\) =(6x1) - (-4 x -3) = 6 - 12 = - 6 ≠ 0

∴ AB is a non-singular matrix

∴ (AB)^-1= \(\frac{1}{|AB|}\) (Adj AB) = \(\frac{1}{-6}\) \(\begin{bmatrix} 1 & 3 \\[0.3em] 4 & 6 \\[0.3em] \end{bmatrix}\)

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Find (AB)-1 if A = \(A = \begin{bmatrix} {1} &amp; {2} &amp; 3 \\[0.3em] {1} &amp; -2 &amp; {-3} \end{bmatrix}\) \(M = \begin{bmatrix} 1 &amp; -1 \\[0.3em] 1 &amp; 2 \\[0.3em] 1 &amp; -2 \end{bmatrix}\)

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Find (AB)-1 if A = \(A = \begin{bmatrix} {1} & {2} & 3 \\[0.3em] {1} & -2 & {-3} \end{bmatrix}\) \(M = \begin{bmatrix} 1 & -1 \\[0.3em] 1 & 2 \\[0.3em] 1 & -2 \end{bmatrix}\)