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## Precalculus

### Course: Precalculus>Unit 7

Lesson 15: Solving linear systems with matrices

# Matrices: FAQ

## What is a matrix?

A matrix is a rectangular array of numbers arranged in rows and columns.
Matrices can be useful for organizing and manipulating data. They can also be used as a tool to help solve systems of equations.

## How do we multiply a matrix by a scalar?

To multiply a matrix by a scalar, we multiply each entry in the matrix by the scalar. For example, if
$A=\left[\begin{array}{cc}2& 3\\ 1& 4\end{array}\right]$
and we want to multiply $A$ by $2$, we can multiply each entry by $2$ to get:
$2A=\left[\begin{array}{cc}4& 6\\ 2& 8\end{array}\right]$

## How do we add or subtract two matrices?

To add or subtract two matrices, we add or subtract corresponding entries. So if
$A=\left[\begin{array}{cc}2& 3\\ 1& 4\end{array}\right]$
and
$B=\left[\begin{array}{cc}5& 2\\ 3& 1\end{array}\right]$,
then
$A+B=\left[\begin{array}{cc}7& 5\\ 4& 5\end{array}\right]$
and
$A-B=\left[\begin{array}{cc}-3& 1\\ -2& 3\end{array}\right]$

## How do we multiply two matrices together?

This one's a bit trickier. To multiply two matrices together, we take the dot product of the rows from the first matrix with the columns from the second matrix. The result is a new matrix. For example, if
$A=\left[\begin{array}{cc}2& 3\\ 1& 4\end{array}\right]$
and
$B=\left[\begin{array}{cc}5& 2\\ 3& 1\end{array}\right]$,
then
$AB=\left[\begin{array}{cc}2\cdot 5+3\cdot 3& 2\cdot 2+3\cdot 1\\ 1\cdot 5+4\cdot 3& 1\cdot 2+4\cdot 1\end{array}\right]=\left[\begin{array}{cc}19& 7\\ 17& 6\end{array}\right].$

## What can we do with the inverse of a matrix?

Finding the inverse of a matrix can be useful for solving linear systems of equations. If $A$ is the matrix representing the system of equations, and $b$ is the vector of solutions, then $Ax=b$. If we can find the inverse of $A$, we can multiply both sides of the equation by it to isolate $x$:
${A}^{-1}Ax={A}^{-1}b⇒x={A}^{-1}b.$

## Where are matrices used in the real world?

Matrices have tons of real-world applications! They can be used in computer graphics to perform transformations on images, in physics to model physical systems, and in statistics to analyze data, just to name a few.