Chapter 4: Q13. (page 199)

Determine whether each pair of matrices are inverses.13.
$X = [\begin{matrix} \frac{1}{3} & - \frac{2}{3} \\ \frac{2}{3} & - \frac{1}{3} \end{matrix}]$
$Y = [\begin{matrix} 1 & 2 \\ 2 & 1 \end{matrix}]$

Short Answer

Expert verified

The given matrices are not inverse of each other.

Step by step solution

- Definition of inverse of matrix.

A square matrixB is said to be an inverse of the square matrixA if $AB = BA = I$ whereL is an identity matrix of the same order as that of matrixA orB .

- Find the product of the given matrices.

The product of the given matrices is:

$\begin{matrix} X \cdot Y = [\begin{matrix} \frac{1}{3} & - \frac{2}{3} \\ \frac{2}{3} & - \frac{1}{3} \end{matrix}] \cdot [\begin{matrix} 1 & 2 \\ 2 & 1 \end{matrix}] \\ = [\begin{matrix} (\frac{1}{3}) (1) + (- \frac{2}{3}) (2) & (\frac{1}{3}) (2) + (- \frac{2}{3}) (1) \\ (\frac{2}{3}) (1) + (- \frac{1}{3}) (2) & (\frac{2}{3}) (2) + (- \frac{1}{3}) (1) \end{matrix}] \\ = [\begin{matrix} \frac{1}{3} + (\frac{- 4}{3}) & \frac{2}{3} + (- \frac{2}{3}) \\ \frac{2}{3} + (- \frac{2}{3}) & \frac{4}{3} + (\frac{- 1}{3}) \end{matrix}] \\ = [\begin{matrix} \frac{1}{3} - \frac{4}{3} & \frac{2}{3} - \frac{2}{3} \\ \frac{2}{3} - \frac{2}{3} & \frac{4}{3} - \frac{1}{3} \end{matrix}] \\ = [\begin{matrix} \frac{1 - 4}{3} & 0 \\ 0 & \frac{4 - 1}{3} \end{matrix}] \\ = [\begin{matrix} \frac{- 3}{3} & 0 \\ 0 & \frac{3}{3} \end{matrix}] \\ = [\begin{matrix} - 1 & 0 \\ 0 & 1 \end{matrix}] \end{matrix}$

As the product of the given matrices is not equal to the identity matrix, therefore the given matrices are not inverse of each other.

- Write the conclusion.

The given matrices are not inverse of each other.

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Determine whether each pair of matrices are inverses.13.
$X = [\begin{matrix} \frac{1}{3} & - \frac{2}{3} \\ \frac{2}{3} & - \frac{1}{3} \end{matrix}]$
$Y = [\begin{matrix} 1 & 2 \\ 2 & 1 \end{matrix}]$

Short Answer

Step by step solution

- Definition of inverse of matrix.

- Find the product of the given matrices.

- Write the conclusion.

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Determine whether each pair of matrices are inverses.13.X=[13-2323-13]Y=[1221]

Short Answer

Step by step solution

­- Definition of inverse of matrix.

­- Find the product of the given matrices.

­- Write the conclusion.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Geometry

Statistics

Probability and Statistics

Calculus

Pure Maths

Study anywhere. Anytime. Across all devices.

Determine whether each pair of matrices are inverses.13.
$X = [\begin{matrix} \frac{1}{3} & - \frac{2}{3} \\ \frac{2}{3} & - \frac{1}{3} \end{matrix}]$
$Y = [\begin{matrix} 1 & 2 \\ 2 & 1 \end{matrix}]$

- Definition of inverse of matrix.

- Find the product of the given matrices.

- Write the conclusion.