Question 16.6.1: (a) Show that the following matrices are inverses of each ot......
(a) Show that the following matrices are inverses of each other:
A=\begin{pmatrix} 5 & 6 \\ 5 & 10 \end{pmatrix} \ \ \ \text{and} \ \ \ X=\begin{pmatrix}\quad \frac{1}{2} & -\frac{3}{10} \\-\frac{1}{4} &\quad \frac{1}{4} \end{pmatrix}(b) Show that the following matrix has no inverse:
A=\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}Step-by-Step
(a) We simply compute, directly, that
A=\begin{pmatrix} 5 & 6 \\ 5 & 10 \end{pmatrix} \begin{pmatrix}\quad \frac{1}{2} & -\frac{3}{10} \\-\frac{1}{4} &\quad \frac{1}{4} \end{pmatrix} =\begin{pmatrix} \frac{5}{2}-\frac{6}{4} & -\frac{15}{10}+\frac{6}{4} \\ \frac{5}{2}-\frac{10}{4} & -\frac{15}{10}+\frac{10}{4} \end{pmatrix} =\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}and likewise we verify that XA = I.
(b) Observe that for all real numbers x, y, z, and w,
Because the element in row 2 and column 2 of the last matrix is 0 and not 1, there is no way of choosing x, y, z, and w to make the product of these two matrices equal to I.
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