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Expert-verified Found in: Page 108 ### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974 # TRUE OR FALSE?There exists a matrix A such that ${\mathbf{A}}\left(\begin{array}{c}\begin{array}{cc}1& 2\\ 1& 2\end{array}\end{array}\right){\mathbf{=}}\left(\begin{array}{cc}1& 1\\ 1& 1\end{array}\right)$ .

The statement is false.

See the step by step solution

## Step 1: Compute the matrix

The matrix is said to be invertible, if and only if the reduced row echelon form of the matrix has nonzero diagonal elements and for second order matrix, the determinant of the matrix should be nonzero.

## Step 2: Apply the application of inverse

Consider the condition.

$A\left(\begin{array}{c}\begin{array}{cc}1& 2\\ 1& 2\end{array}\end{array}\right)=\left(\begin{array}{cc}1& 1\\ 1& 1\end{array}\right)\phantom{\rule{0ex}{0ex}}⇒A=\left(\begin{array}{cc}1& 1\\ 1& 1\end{array}\right){\left(\begin{array}{c}\begin{array}{cc}1& 2\\ 1& 2\end{array}\end{array}\right)}^{-1}$

If the determinant of the matrix is zero, then, this inverse does not exist.

As $det\left(\begin{array}{cc}1& 2\\ 1& 2\end{array}\right)=0$ thus, the inverse does not exist.

Therefore, the matrix does not exist.

The statement is false. ### Want to see more solutions like these? 