Question

TRUE OR FALSE

25. $\left[\begin{array}{cc}3& -4\\ 4& 3\end{array}\right]$ is an orthogonal matrix.

Short answer

The given statement is false.

Step-by-step solution

Step-by-step solution Step 1: Determine the definition of an orthogonal matrix

A square matrix with real numbers or elements is said to be an orthogonal matrix if its transpose is equal to its inverse matrix.

Or it can be said that, when the product of a square matrix and its transpose gives an identity matrix, then the square matrix is known as an orthogonal matrix.

Consider an example

For example,

Suppose A is a square matrix with real elements and of $n\times n$order and $A^T$ is the transpose of A. Then according to the definition, if $A^T=A^{-1}$ is satisfied, then,

$A·A^T=I$

Consider the matrix below:

$\left[\begin{array}{cc}3& -4\\ 4& 3\end{array}\right]$

The column vectors are no unit vectors in this matrix.

Thus, the matrix will not be an orthogonal matrix.

Hence, the given statement is false.