Topic 8 of 8

Introduction to matrices

Step 13 · Algebra 2 topics

Add, subtract and scale matrices, multiply them, and find the determinant and inverse of a 2 × 2 matrix to solve a system of equations.

Adding and scaling matrices

A matrix is a rectangle of numbers in rows and columns. A 2 × 3 matrix has 2 rows and 3 columns. Matrices of the same size add entry by entry, and multiplying by a number multiplies every entry.

\begin{pmatrix} 1 & 4 \\ -2 & 0 \end{pmatrix} + \begin{pmatrix} 3 & -1 \\ 5 & 2 \end{pmatrix} = \begin{pmatrix} 4 & 3 \\ 3 & 2 \end{pmatrix}

Example: Scaling

Find 3A, where $A = \begin{pmatrix} 2 & -1 \\ 0 & 4 \end{pmatrix}$ .

Multiply every entry by 3.

Answer $\begin{pmatrix} 6 & -3 \\ 0 & 12 \end{pmatrix}$

Matrices store data neatly: the rows could be stores, the columns products, and each entry the number sold.

Multiplying matrices

To multiply two matrices, each entry of the answer is a row of the first times a column of the second: multiply the matching entries and add.

\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} = \begin{pmatrix} 19 & 22 \\ 43 & 50 \end{pmatrix}

Example: One entry

In the product above, how is 43 found?

Row 2 of the first matrix, (3, 4), times column 1 of the second, (5, 7).
3 × 5 + 4 × 7 = 15 + 28 = 43.

Answer43

Example: Sizes

A is 2 × 3 and B is 3 × 4. What size is AB? Does BA exist?

For AB the inner sizes, 3 and 3, match, so AB is 2 × 4.
For BA the inner sizes are 4 and 2, which do not match.

AnswerAB is 2 × 4, and BA does not exist

Determinants and inverses

The determinant of a 2 × 2 matrix is ad − bc. If it is not 0, the matrix has an inverse, which undoes it.

A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \quad \det A = ad - bc, \quad A^{-1} = \frac{1}{ad - bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}

Example: A determinant

Find the determinant of $\begin{pmatrix} 4 & 3 \\ 2 & 5 \end{pmatrix}$ .

4 × 5 − 3 × 2 = 20 − 6 = 14.

Answer14

Example: An inverse

Find the inverse of $\begin{pmatrix} 3 & 1 \\ 5 & 2 \end{pmatrix}$ .

The determinant is 3 × 2 − 1 × 5 = 1.
Swap a and d, and change the signs of b and c.

Answer $\begin{pmatrix} 2 & -1 \\ -5 & 3 \end{pmatrix}$

Example: Solving a system

Solve 3x + y = 5 and 5x + 2y = 9 with the inverse.

The system is $\begin{pmatrix} 3 & 1 \\ 5 & 2 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 5 \\ 9 \end{pmatrix}$ .
Multiply by the inverse: x = 2 × 5 − 1 × 9 = 1 and y = −5 × 5 + 3 × 9 = 2.

Answerx = 1, y = 2

Try it yourself

Work each one on paper first, then open the answer to check.

What size is a matrix with 3 rows and 2 columns?
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3 × 2
Add $\begin{pmatrix} 2 & 0 \\ 1 & 3 \end{pmatrix} + \begin{pmatrix} -1 & 4 \\ 2 & 2 \end{pmatrix}$ .
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$\begin{pmatrix} 1 & 4 \\ 3 & 5 \end{pmatrix}$
A is 3 × 2 and B is 2 × 5. What size is AB?
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3 × 5
Multiply $\begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix}\begin{pmatrix} 3 \\ 4 \end{pmatrix}$ .
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$\begin{pmatrix} 3 \\ 10 \end{pmatrix}$
Find the determinant of $\begin{pmatrix} 6 & 2 \\ 3 & 1 \end{pmatrix}$ .
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0
Find the determinant of $\begin{pmatrix} 2 & -3 \\ 4 & 1 \end{pmatrix}$ .
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14