If A = ((3,-4),(-1,2)) and B is a square matrix of order 2 such that AB=I then B = ? ​

Question

If A = \(\begin{pmatrix} 3&-4 \\ - 1 &2 \end{pmatrix}\) and B is a square matrix of order 2 such that AB=I then B = ?

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

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

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

D. None of these

Answer 1

B = I 

B = A^-1 I ---------1 

A^-1= 1/|A| adj A --------- 2 

|A| = 3 x 2 – (-4) x (-1)

= 2 

C₁₁ = 2 C₁₂ = 1 

C₂₁ = 4 C₂₂ = 3 

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

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

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

Putting in 2

A^-1 = 1/|2| \(\begin{pmatrix}2&4 \\ 1 &3 \end{pmatrix}\)

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

Putting in 1 

B = A^-1 I

= A^-1

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