Matrices

In this section we will learn to multiply a matrix by a scalar, solve matrix equations, and add, subtract and multiply matrices

Definition of a Matrix: A matrix is a rectangular arrangement or array of numbers or functions, in the form of horizontal and vertical lines and subject to certain rules of operations.

Matrices are usually denoted by capital letters of the alphabet.

Very often capital letters A, B, C, ... are used to denote a matrix.

If mn numbers or functions are arranged in the form of a rectangular array Z, having m rows and n columns, then Z is called a m × n matrix.

Matrices (the plural of matrix) can be multiplied by a number - similar to distributive property. They can be added, subtracted, and multiplied. You can solve equations using matrices.

When describing a matrix, it is always row by column (row x column) . This would be called a 4x5 matrix.

An element in the matrix is identified by its position.

This is a 3x2 matrix

The Zero Matrix is the identity element for addition. Any matrix added to it would yield the exact same matrix.

Any matrix multiplied by by the Identity matrix would yield the exact same matrix

Addition of Matrices

In order to add matrices, they must have the same number of rows and same number of columns.

When adding, add the element is the same position of each matrix. In other words, given matrices A and B. The element in row 1, column 1 of matrix A would be added to the element in row 1, column 1 of matrix B.

Subtraction of matrices would follow the same process.

Multiplication of Matrices

The first type of matrix multiplication is easy - it is similar to distributive property, where each element in the matrix is multiplied by the scalar (the number outside the matrix.)

Notice this is similar to distributive property

Multiplication of matrices is much more difficult than addition with more steps involved.

First: You must check the dimensions of the matrices. In order to multiply matrix A and matrix B, the number of COLUMNS in matrix A must match the number of ROWS in matrix B. If matrix A was 3x5 and matrix B was 5x2, they could be added. But, if matrix A was 3x5 and matrix B was 3x2, then they could not be added. When multiplying a 3x2 and a 2x5 (which does have the columns of the first matrix matching the rows of the second matrix) the result would be a 3x5 matrix. If multiplying a 2x8 and 8x4, the result would be a 2x4 matrix

Multiplying 2 matrices is different. The entire first row multiplies the entire first column giving you ONE element.

Here, the first row in matrix A (a11 and a12) will multiply the first column in matrix B (b11 and b21) and the sum yields ONE element in the solution matrix - the element in the first row , first column

EXAMPLE 1

EXAMPLE 2

EXAMPLE 3

EXAMPLE 4

Here, we have a 1x3 matrix (where the columns the type of pies) multiplying a 3x4 matrix (where the rows represent pies). Remember from before, the COLUMNS of the first matrix must match the ROWS of the second matrix and the result would be a 1x4 matrix.

Practice with Matrices

Practice with Matrices 2

Now let's try some word problems

Think about what the two matrices would be. Remember that the number of columns for the first matrix must match the number of rows for the second matrix in order to multiply.

X = 5, y = 3, z = -2

Time to Practice

ANSWERS:

End of Unit Test with Solutions

Practice with Answers

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