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.

## Algebra (all content)

### Unit 20: Lesson 1

Introduction to matrices

# Intro to matrices

Matrix is an arrangement of numbers into rows and columns. Make your first introduction with matrices and learn about their dimensions and elements.
A matrix is a rectangular arrangement of numbers into rows and columns.
For example, matrix A has two rows and three columns.

## Matrix dimensions

The dimensions of a matrix tells its size: the number of rows and columns of the matrix, in that order.
Since matrix A has start color #11accd, start text, t, w, o, space, r, o, w, s, end text, end color #11accd and start color #e07d10, start text, t, h, r, e, e, space, c, o, l, u, m, n, s, end text, end color #e07d10, we write its dimensions as start color #11accd, 2, end color #11accd, times, start color #e07d10, 3, end color #e07d10, pronounced "two by three".
In contrast, matrix B has start color #11accd, start text, t, h, r, e, e, space, r, o, w, s, end text, end color #11accd and start color #e07d10, start text, t, w, o, space, c, o, l, u, m, n, s, end text, end color #e07d10, so it is a start color #11accd, 3, end color #11accd, times, start color #e07d10, 2, end color #e07d10 matrix.
$B=\left[\begin{array}{rr}-8 & -4 \\ 23 & 12 \\18 &10 \end{array}\right]$
When working with matrix dimensions, remember start color #11accd, start text, r, o, w, s, end text, end color #11accd, times, start color #e07d10, start text, c, o, l, u, m, n, s, end text, end color #e07d10!

1) What are the dimensions of matrix D?
$D=\left[\begin{array}{rr}-7 & 24 &~~~~2 \\ 1 & 15 &~~11 \\-9 &12&~~~0 \\8 &-3&-1 \end{array}\right]$
times

2) What are the dimensions of matrix E?
$E=\left[\begin{array}{rr}-2 & 6 &1 &3 \\ 0& -8 &3&10 \end{array}\right]$
times

3) What are the dimensions of matrix F?
$F=\left[\begin{array}{rr}-2 \\ 0& \\10 \end{array}\right]$
times

## Matrix elements

A matrix element is simply a matrix entry. Each element in a matrix is identified by naming the row and column in which it appears.
For example, consider matrix G:
$G=\left[\begin{array}{rr}4& 14&-7 \\ 18 & 5&13 \\-20 &4&22 \end{array}\right]$
The element g, start subscript, start color #11accd, 2, end color #11accd, comma, start color #e07d10, 1, end color #e07d10, end subscript is the entry in the start color #11accd, start text, s, e, c, o, n, d, space, r, o, w, end text, end color #11accd and the start color #e07d10, start text, f, i, r, s, t, space, c, o, l, u, m, n, end text, end color #e07d10.
In this case g, start subscript, 2, comma, 1, end subscript, equals, start color #ca337c, 18, end color #ca337c.
In general, the element in start color #11accd, start text, r, o, w, space, end text, i, end color #11accd and start color #e07d10, start text, c, o, l, u, m, n, space, end text, j, end color #e07d10 of matrix A is denoted as a, start subscript, start color #11accd, i, end color #11accd, comma, start color #e07d10, j, end color #e07d10, end subscript.

4) $A=\left[\begin{array}{rr}2 & -4 &~~~8 \\ 1 & 5&-5 \\-2 &6&~~~2 \end{array}\right]$
a, start subscript, 1, comma, 3, end subscript, equals

5) $B=\left[\begin{array}{rr}12 & -2.3 \\ 4.6 & 1.2& \end{array}\right]$
b, start subscript, 2, comma, 1, end subscript, equals

6) Matrix C is a 2, times, 3 matrix with c, start subscript, 1, comma, 2, end subscript, equals, 6.
Which could be matrix C?

## Want to join the conversation?

• On problem six why doesn't answer B not satisfy the equation? But answer C does? The six is in the same spot as in both answers.
• To satisfy all the conditions, there must be a 2 x 3 matirce. Both C a d D satisfy the first condition, then the second conditon is that the value of row 1, entry 2 be a 6. Only C satisfies that condition. Therefore that is the correct matrix.
• on problem 6, why is the answer c and not b ? it is the same answer just in different positions.
• No, since it is row by columns, answer b would be a 3 x 2 matrix not a 2 x 3 matrix, hope this helps
• why is there two answers to question number 6?
• #2 is wrong because it is a 3x2 matrix. We always count rows first, then columns. How tall is the matrix (2) then how wide is it (3)? That leaves #3 and #4 as options. Row 1 column 2 = 6. That means #3 is the right choice.
• Hi my name is Sayan, I still haven't understood the relevance of matrices. My request to you would be if you could give a real life scenario where matrices are used and/or another place where it has a practical use in everyday life. Also do explain this with an example for my better understanding?
• I'm sure you can carry out everyday activities like buying groceries or make a video call to your grandma without manually calculating matrices.

However, matrices are absolutely necessary for creating mathematical models of complex real-world phenomena, crunching numbers at massive scales, producing any kind of computer of computer graphics, and a lot more therein.

I don't know about your everyday life, but millions of people are employing matrix math in their everyday lives professionally to make possible all of the niceties we enjoy in 21st century life, such as the aforementioned grocery store and video chat with grandma.
• Can there be a matrix with 0x0. If yes, then how is it represented
• It could be just an empty matrix, like this: []
However, such a matrix could not contain any information and would therefore be useless.
• What are matrices used for in math?
• A lot of things can be thought of as transformations of space, taking every point in 3D space and moving it somewhere else. Matrices are a compact way of talking about and working with a certain class of transformations.
• Is [0] the same as an empty one []?
• Probably no, because [] is empty, but [0] isn't. They're called zero matrices and they're used in matrices the same way regular zeroes are.