Matrix Addition and Multiplication
The matrices
\[\mathbf{A} = \renewcommand{\arraystretch}{1.5} \begin{pmatrix} 1 & x \\ y & -\frac{9}{5} \end{pmatrix} \renewcommand{\arraystretch}{1} \quad \text{and} \quad \mathbf{B} = \renewcommand{\arraystretch}{1.5} \begin{pmatrix} \frac{12}{5} & \frac{1}{10} \\ 5 & z \end{pmatrix} \renewcommand{\arraystretch}{1}\]satisfy $ \mathbf{A} + \mathbf{B} = \mathbf{A} \mathbf{B} $. Find $ x + y + z $.
- 1
- 2
- 3
- +
- 4
- 5
- 6
- -
- 7
- 8
- 9
- $\frac{a}{b}$
- .
- 0
- =
- %
- $a^n$
- $a^{\circ}$
- $a_n$
- $\sqrt{}$
- $\sqrt[n]{}$
- $\pi$
- $\ln{}$
- $\log$
- $\theta$
- $\sin{}$
- $\cos{}$
- $\tan{}$
- $($
- $)$
- $[$
- $]$
- $\cap$
- $\cup$
- $,$
- $\infty$
Solution
We have that
\[\mathbf{A} + \mathbf{B} = \renewcommand{\arraystretch}{1.5} \begin{pmatrix} \frac{17}{5} & x + \frac{1}{10} \\ y + 5 & z - \frac{9}{5} \end{pmatrix} \renewcommand{\arraystretch}{1}\]and
\[\mathbf{A} \mathbf{B} =
\renewcommand{\arraystretch}{1.5} \begin{pmatrix} 1 & x \\ y & -\frac{9}{5} \end{pmatrix} \renewcommand{\arraystretch}{1}
\renewcommand{\arraystretch}{1.5} \begin{pmatrix} \frac{12}{5} & \frac{1}{10} \\ 5 & z \end{pmatrix} \renewcommand{\arraystretch}{1}
=
\renewcommand{\arraystretch}{1.5} \begin{pmatrix} 5x + \frac{12}{5} & xz + \frac{1}{10} \\ \frac{12}{5} y - 9 & \frac{1}{10} y - \frac{9}{5} z \end{pmatrix} \renewcommand{\arraystretch}{1}
.\]Thus,
\begin{align*}
5x + \frac{12}{5} &= \frac{17}{5}, \\
xz + \frac{1}{10} &= x + \frac{1}{10}, \\
\frac{12}{5} y - 9 &= y + 5, \\
\frac{1}{10} y - \frac{9}{5} z &= z - \frac{9}{5}.\end{align*}From the first equation, $ x = \frac{1}{5}, $ and from the third equation, $ y = 10 $. Then from either the second equation or fourth equation, $ z = 1 $. Hence, $ x + y + z = \frac{1}{5} + 10 + 1 = \boxed{\frac{56}{5}} $.
Freeflows