If A = ((2,-1),(1,3)), then A^{-1} = ? ​

Question

If A = \( \begin{pmatrix} 2&-1 \\ 1 &3 \end{pmatrix}\), then A^-1 = ?

A.\( \begin{pmatrix} 3/7&-1/7 \\ 1/7 &2/7 \end{pmatrix}\)

B.\( \begin{pmatrix} 3/7&1/7 \\ -1/7 &2/7 \end{pmatrix}\)

C.\( \begin{pmatrix} 1/3&1/7 \\ 1/7 &2/7 \end{pmatrix}\)

D. None of these

Answer 1

1 = 1/|A| adj A --------- 1 

|A|  = 3 x 2 – (1) x (-1) 

= 7 

C₁₁ = 3 C₁₂ = -1 

C₂₁ = 1 C₂₂ = 2

Cofactor matrix A = \( \begin{pmatrix} 2&1 \\ 4 &3 \end{pmatrix}\)

Adj A = \( \begin{pmatrix} 3&-1 \\ 1 &2 \end{pmatrix}\)

= \( \begin{pmatrix} 3&1 \\ -1 &2 \end{pmatrix}\)

Putting in 1

A^-1 = 1/|7| \( \begin{pmatrix} 3&1 \\ -1 &2 \end{pmatrix}\)

= \( \begin{pmatrix} 3/7&1/7 \\ -1/7 &2/7 \end{pmatrix}\)