Matrix fun

Note: This is only a lengthy response because I explained how to do each step, but there's only three steps to solving it.

Step 1: Solve for $AB$ and $A+B$

$A * B = \begin{bmatrix} (1*2.4) + 5x & (1*0.1) + xz \\ 2.4y + (-1.8*5) & 0.1y - 1.8z \\ \end{bmatrix}$ $= \begin{bmatrix} 2.4+5x & 0.1+xz \\ 2.4y-9 & 0.1y-1.8z \\ \end{bmatrix}$

$A + B = \begin{bmatrix} 2.4+1 & 0.1+x \\ 5+y & z - 1.8 \\ \end{bmatrix}$ $= \begin{bmatrix} 3.4 & 0.1+x \\ 5+y & z - 1.8 \\ \end{bmatrix}$

Step 2: Set $AB$ equal to $A+B$

So, now that we've done the math, we can equate $AB$ and $A+B$ in terms of $x, y, z$ .

$\begin{bmatrix} 3.4 & 0.1+x \\ 5+y & z - 1.8 \\ \end{bmatrix}$ $= \begin{bmatrix} 2.4+5x & 0.1+xz \\ 2.4y-9 & 0.1y-1.8z \\ \end{bmatrix}$

I've written the equations to solve below in case this is easier to understand:

$\begin{bmatrix} 3.4=2.4+5x & 0.1+x=0.1+xz \\ 5+y=2.4y-9 & z-1.8=0.1y-1.8z \\ \end{bmatrix}$

All that's left to do here is solve for the variables. There are only two equations that depend only on one variable, so we'll start with those first:

$3.4=2.4 + 5x$

$\rightarrow x=\frac{1}{5}$

$5+y=2.4y-9 \rightarrow 14 = 1.4y$

$\rightarrow y=10$

So now that we know $x$ and $y$ , we can plug these into the remaining two equations to solve for $z$ , which will give $z=1$ .

Step 3: Add $x+y+z$

$10+\frac{1}{5}+1=11.2$

So, the answer is 11.2

Writing out the multiplication and addition, $\left[\begin{array}{cc} 2.4 + 5x & 0.1 + xz \\ 2.4y - 9 & 0.1y - 1.8z \end{array}\right] = \left[\begin{array}{cc} 3.4 & 0.1 + x \\ y + 5 & -1.8 + z \end{array}\right].$ From the top-left element we see that $x = 0.2$ ; from the top-right element that $z = 1$ ; from the bottom-right element that $y = 10$ .