Find the matrix A such that A = [(2,3)(4,5)] = [(0,-4)(10,3)].

Question

Find the matrix A such that A. \( \begin{bmatrix} 2 & 3 \\[0.3em] 4& 5 \\[0.3em] \end{bmatrix}\) = \( \begin{bmatrix} 0& -4 \\[0.3em] 10& 3 \\[0.3em] \end{bmatrix}.\) 

A = [(2,3)(4,5)] = [(0,-4)(10,3)].

Answer 1

Given : A. \( \begin{bmatrix} 2& 3 \\[0.3em] 4& 5 \\[0.3em] \end{bmatrix}\) = \( \begin{bmatrix} 0& -4 \\[0.3em] 10& 3 \\[0.3em] \end{bmatrix}.\) 

To find : matrix A

Formula used :

Where c_ij = a_i1b_1j + a_i2b_2j + a_i3b_3j + ……………… + a_inb_nj

IF AX = B, then A = BX -1

To find \( \begin{bmatrix} 2 & 3 \\[0.3em] 4 & 5 \\[0.3em] \end{bmatrix}^{-1}\)

Determinant of given matrix = \( \begin{bmatrix} 2 & 3 \\[0.3em] 4 & 5 \\[0.3em] \end{bmatrix}\) = 5(2) – (4)(3) = 10 – 12 = -2

Adjoint of matrix \( \begin{bmatrix} 2 & 3 \\[0.3em] 4 & 5 \\[0.3em] \end{bmatrix}\) = \( \begin{bmatrix} 5&-3 \\[0.3em] -4&2 \\[0.3em] \end{bmatrix}\) 

A = \( \begin{bmatrix} -8 & 4 \\[0.3em] -19&12 \\[0.3em] \end{bmatrix}\)

A = \( \begin{bmatrix} -8 & 4 \\[0.3em] -19&12 \\[0.3em] \end{bmatrix}\)