Question

If $5A=\left(\begin{array}{cc}3& -4\\ 4& x\end{array}\right)$ and $A{A}^T=A^{T}A=I$ then $x=$

• A. $3$
• B. $-3$
• C. $2$
• D. $-2$

Answer 1

The correct option is A $3$

Given,

$5A=\left(\begin{array}{cc}3& -4\\ 4& x\end{array}\right)$

or, $A=\frac{1}{5}\left(\begin{array}{cc}3& -4\\ 4& x\end{array}\right)$.

Also given, $A{A}^T=A^{T}A=I$, this is the definition of the orthogonal matrix.

We know for an orthogonal matrix $|A|=\pm 1$.

Then we have,

$\frac{1}{25}(3x+16)=\pm 1$ [ Since the order of the matrix is $2$]

or, $3x+16=\pm 25$

or, $3x=25-16$ [ Taking the positive value]

or, $3x=9$

or, $x=3$.