I am trying to figure out the below question:
15. For which values of the constants $a$ and $b$ is the matrix
$$A = \left[\begin{array}{cc} a & -b \\ b & a \end{array}\right]$$
invertible? What is the inverse in this case? See Exercise 13.
My understanding is that a matrix is invertible when the determinant is not zero. In this case, when $a^2 – b^2 = 0$ the matrix is not invertible. Thus, for any values $a,b$ such that $a^2$ does not equal $b^2$, the matrix is invertible.
However, the solutions in the back of the book state that the matrix is invertible if $a$ does not equal zero or if $b$ does not equal zero. Can someone explain this to me?
Best Answer
The determinant of the matrix is $a^2 + b^2$, not $a^2 - b^2$. This is non-zero if and only if at least one of $a$ and $b$ is non-zero.
This assumes that $a$ and $b$ are real; otherwise, the issue is more complicated. For example, the matrix
$$\left[\begin{array}{cc} i & -1 \\ 1 & i\end{array}\right]$$ is not invertible.