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Mathematics Question on Matrices

Find the inverse of each of the matrices, if it exists. [2−6 1−2]\begin{bmatrix} 2 & -6 \\\ 1 & -2 \end{bmatrix}

Answer

Let A=[2−6 1−2]\begin{bmatrix} 2 & -6 \\\ 1 & -2 \end{bmatrix}

We know that A=AIA = AI

[2−6 1−2]\begin{bmatrix} 2 & -6 \\\ 1 & -2 \end{bmatrix}=A[10 01]\begin{bmatrix} 1 & 0 \\\ 0 & 1 \end{bmatrix}

⇒ [20 11]\begin{bmatrix} 2 & 0 \\\ 1 & 1 \end{bmatrix}=A [13 01]\begin{bmatrix} 1 & 3 \\\ 0 & 1 \end{bmatrix} (C2→C2+3C1)(C_2\rightarrow C_2+3C_1)

⇒ [20 11]\begin{bmatrix} 2 & 0 \\\ 1 & 1 \end{bmatrix} A [−23 −11]\begin{bmatrix} -2 & 3 \\\ -1 & 1 \end{bmatrix} (C1→C1−C2)(C_1\rightarrow C_1-C_2)

⇒ [10 01]\begin{bmatrix} 1 & 0 \\\ 0 & 1 \end{bmatrix} =A [−13 −121]\begin{bmatrix} -1 & 3 \\\ -\frac12 & 1 \end{bmatrix} (C1→12C1)(C_1\rightarrow \frac{1}{2}C_1)

∴\thereforeA-1=[−13 −121]\begin{bmatrix} -1 & 3 \\\ -\frac12 & 1 \end{bmatrix}