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Question

Mathematics Question on Matrices

Find the inverse of each of the matrices, if it exists. [2−3 −12]\begin{bmatrix} 2 & -3\\\ -1 & 2 \end{bmatrix}

Answer

Let A=[2−3 −12]\begin{bmatrix} 2 & -3\\\ -1 & 2 \end{bmatrix}

We know that A = IA

[2−3 −12]\begin{bmatrix} 2 & -3\\\ -1 & 2 \end{bmatrix}=A[10 01]\begin{bmatrix} 1 & 0\\\ 0 & 1 \end{bmatrix}

⇒ [1−1 −12]\begin{bmatrix} 1 & -1\\\ -1 & 2 \end{bmatrix}= [11 01]\begin{bmatrix} 1 & 1\\\ 0 & 1 \end{bmatrix}A (R1→R1+R2)(R_1\rightarrow R_1+R_2)

⇒ [1−1 01]\begin{bmatrix} 1 & -1\\\ 0 & 1 \end{bmatrix}=[11 12]\begin{bmatrix} 1 & 1\\\ 1 & 2 \end{bmatrix}A (R2→R2+R1)(R_2\rightarrow R_2+R_1)

⇒ [10 01]\begin{bmatrix} 1 & 0\\\ 0 & 1 \end{bmatrix}= [23 12]\begin{bmatrix} 2 & 3\\\ 1 & 2 \end{bmatrix}A (R1→R1+R2)(R_1\rightarrow R_1+R_2)

so A-1=[23 12]\begin{bmatrix} 2 & 3\\\ 1 & 2 \end{bmatrix}