# Solving for h and w in 2 non-invertible matrices

PLEASE NOTE: I am not asking for you to solve the problem, but tell me the steps to arrive at the solution (though leaving the answer would help)

In my Linear Algebra class, we have been given the question:

Given the following non-invertible matrices, solve for $$h$$ and $$w$$, where:

$$H = \begin{pmatrix} 4h - w + 2 & 0 \\ -8 & 1 \end{pmatrix}, \hspace{1cm} W = \begin{pmatrix} 3 & 10 \\ 0 & 2h + 5w + 1 \end{pmatrix}.$$

I am not sure how to begin solving this problem, and any help would be much appreciated.

Also, if anyone knows the proper name of this process, that would also be helpful.

• If I understand the question correct, you shall calculate $(h, w)$ such that $H$ and $W$ are not invertible. This is e. g. the case if the determinants of $H$ and $W$ are zero, which will lead you to an homogenous linear equation system. – Jan Jun 2 at 20:04

Hint: You are told that your two matrices are not invertible. Thus their determinants are equal to $$0$$. Try actually computing the determinants to get a pair of linear equations (each of which will be of the form $$Ah + Bw + C = 0$$ for suitable constants $$A,B,C$$). Now you just have to solve two linear equations in two unknowns.
Use the fact that non-invertible matrices have $$0$$ determinant. You should get $$w=0,h=-1/2$$.