Question

Question: If A is a non-zero matrix of the form,$\left[\begin{array}{cc}a& -b\\ b& a\end{array}\right]$ then the rank of A must be 2.

Short answer

Answer:

True, If A is a non-zero matrix of the form, $\left[\begin{array}{cc}a& -b\\ b& a\end{array}\right]$then the rank of A is 2.

Step-by-step solution

Rank of a matrix

For the matrix, if we need to find the rank of the matrix, we will find the determinant of a matrix.

We have.$A=\left[\begin{array}{cc}a& -b\\ b& a\end{array}\right]\Rightarrow \left|A\right|=\left|a^2+b^2\right|$

Justification of answer

Now, since the value of a and b can’t be zero at the same time. Because A is a non-zero matrix.

$\Rightarrow \left|a^2+b^2\right|\ne 0$

Thus, the rank of the matrix will be 2.

Hence, it is true thatIf A is a non-zero matrix of the form,$\left[\begin{array}{cc}a& -b\\ b& a\end{array}\right]$then the rank of A is 2.