If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

## 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.

## Want to join the conversation?

• this unit broke me
• ChatGPT is good to communicate with, not get answers but talk about topics and theorems and such and reiterate it all in your own words and allow it to check your logic aswell.
• How is The Matrix, the movie relevent to matrices?
• Because in the video in Algebra on matrices Sal said that he thinks the robots in the movie used the formula for matrices.
• what is gaussian elimination?
(1 vote)
• Gaussian elimination is a method of manipulating the rows of a matrix representing a system of linear equations, in order to solve the system.

Permissible row operations include adding a multiple of one row to another row, dividing a row by a nonzero number, and switching the order of the rows.

The goal is to get the matrix into a form called row reduced echelon form, so that the solution set can be easily read from the matrix.

Have a blessed, wonderful day!