Grade 12

Advanced

6 min

Lesson 8 of 12

Matrix Operations and Properties

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Introduction to linear algebra through matrix addition, multiplication, and scalar transformations.

How does a computer instantly rotate a 3D character in a video game or filter a photo on Instagram? The secret lies in matrices—mathematical grids that act as the engine for almost every modern technology.

Learning Objectives

1.Perform addition, subtraction, and scalar multiplication on matrices of various dimensions
2.Execute matrix multiplication and identify when the operation is defined
3.Analyze the properties of identity and zero matrices in linear systems

The Basics: Addition and Scaling

A matrix is a rectangular array of numbers arranged in rows and columns. To perform addition or subtraction, the matrices must have the same dimensions ( $m \times n$ ). You simply add or subtract the corresponding elements. Scalar multiplication involves multiplying every single element in the matrix by a real number (the scalar). Think of this as 'stretching' or 'shrinking' the entire data set. If matrix $A$ represents a set of coordinates, $2A$ doubles the distance of every point from the origin.

Given $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 0 & 5 \\ -1 & 2 \end{pmatrix}$ , find $2A + B$ .

1. Multiply $A$ by the scalar 2: $2A = \begin{pmatrix} 2(1) & 2(2) \\ 2(3) & 2(4) \end{pmatrix} = \begin{pmatrix} 2 & 4 \\ 6 & 8 \end{pmatrix}$ . 2. Add the result to $B$ : $\begin{pmatrix} 2+0 & 4+5 \\ 6+(-1) & 8+2 \end{pmatrix} = \begin{pmatrix} 2 & 9 \\ 5 & 10 \end{pmatrix}$ .

Quick Check

If Matrix A is $3 \times 2$ and Matrix B is $2 \times 3$ , can you add them together?

Check the number of rows and columns for both.

Answer

No, matrices must have the exact same dimensions to be added or subtracted.

The Power of Multiplication

Matrix multiplication is not element-wise; it is a dot product of rows and columns. For the product $AB$ to exist, the number of columns in $A$ must equal the number of rows in $B$ . If $A$ is $m \times n$ and $B$ is $n \times p$ , the resulting matrix $C$ will be $m \times p$ . This operation is the foundation of linear transformations, allowing us to calculate complex movements in space by combining simpler ones.

Multiply $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 5 \\ 6 \end{pmatrix}$ .

1. Check dimensions: $A$ is $2 \times 2$ , $B$ is $2 \times 1$ . The inner numbers (2 and 2) match. Result will be $2 \times 1$ . 2. Calculate the first row: $(1 \times 5) + (2 \times 6) = 5 + 12 = 17$ . 3. Calculate the second row: $(3 \times 5) + (4 \times 6) = 15 + 24 = 39$ . 4. Result: $\begin{pmatrix} 17 \\ 39 \end{pmatrix}$ .

Quick Check

If you multiply a $4 \times 3$ matrix by a $3 \times 5$ matrix, what are the dimensions of the resulting matrix?

The outer dimensions of the two matrices determine the size of the product.

Answer

The resulting matrix will be $4 \times 5$ .

Special Matrices and the Rule-Breakers

In standard algebra, $a \times b = b \times a$ . In linear algebra, matrix multiplication is NOT commutative; usually, $AB \neq BA$ . However, we have special matrices that behave like numbers. The Identity Matrix ( $I$ ), a square matrix with 1s on the main diagonal and 0s elsewhere, acts like the number 1 ( $AI = A$ ). The Zero Matrix ( $O$ ), filled entirely with 0s, acts like the number 0 ( $A + O = A$ ).

Let $A = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ and $B = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ .

1. Calculate $AB$ : $\begin{pmatrix} (1\times0 + 0\times0) & (1\times1 + 0\times0) \\ (0\times0 + 0\times0) & (0\times1 + 0\times0) \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ . 2. Calculate $BA$ : $\begin{pmatrix} (0\times1 + 1\times0) & (0\times0 + 1\times0) \\ (0\times1 + 0\times0) & (0\times0 + 0\times0) \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$ . 3. Since $AB \neq BA$ , order matters!

Key Takeaways

1Addition and subtraction require identical dimensions ( $m \times n$ ).
2Matrix multiplication $AB$ is defined only if the number of columns in $A$ equals the rows in $B$ .
3The Identity Matrix ( $I$ ) leaves a matrix unchanged during multiplication ( $AI = A$ ).
4Matrix multiplication is generally non-commutative ( $AB \neq BA$ ).

Quick Check

Multiple Choice

What is the result of $3 \times \begin{pmatrix} 2 & -1 \\ 0 & 4 \end{pmatrix}$ ?

Multiple Choice

If matrix $M$ is $5 \times 2$ and matrix $N$ is $2 \times 8$ , what is the dimension of $MN$ ?

True/False

For any two square matrices $A$ and $B$ of the same size, $AB$ always equals $BA$ .

What's Next?

Review Tomorrow

In 24 hours, try to sketch a $2 \times 2$ Identity matrix and explain why multiplying it by another matrix doesn't change the values.

Practice Activity

Find a simple $2 \times 2$ matrix and calculate its square ( $A^2 = A \times A$ ). Then, try to find a matrix where $A^2$ results in the Zero Matrix.

Browse

Browse

Matrix Operations and Properties

Learning Objectives

The Basics: Addition and Scaling

The Power of Multiplication

Special Matrices and the Rule-Breakers

Key Takeaways

Quick Check

What's Next?

Matrix Operations and Properties

Learning Objectives

The Basics: Addition and Scaling

The Power of Multiplication

Special Matrices and the Rule-Breakers

Key Takeaways

Quick Check

What's Next?