Which coefficients make the matrix $A$ singular

Question

I am working on this exercise:

Which numbers a, b make the matrix singular?

the matrix is the following:

$\begin{bmatrix}1 & 2 & 0\\a & 8 & 3\\0 & b & 5\end{bmatrix}$

I know that I need to find values for $a$ and $b$ which ensure that all the rows are linearly independent. However I am unsure on how to proceed.

Answer 1

Calculate the determinant of $A$ and find the values of $a,b$ that make the determinant equal to $0$ $$\det(A)=\begin{vmatrix}1&2&0\\a&8&3\\0&b&5\end{vmatrix}=1(40-3b)-2(5a-0)=40-3b-10a\overset{!}=0$$ which gives $$b=\frac13(10a-40)$$ So, the matrix $A$ is singular for all pairs $a \in\mathbb R, b=\frac{10}3(a-4)$.

Answer 2

A matrix is singular if and only if its determinant is 0. Calculating the determinant of this matrix, we get a linear equation in the $a, b$. Solve that.

Answer 3

You may just write $$ \begin{vmatrix}1 & 2 & 0\\a & 8 & 3\\0 & b & 5\end{vmatrix}=0 $$ that is $$ \begin{vmatrix}1 & 2 & 0\\a & 8 & 3\\0 & b & 5\end{vmatrix}=1\times \begin{vmatrix}8 & 3\\b & 5\end{vmatrix}-2 \times\begin{vmatrix}a & 3\\0 & 5\end{vmatrix}=40-3b-10a=0 $$