7.3 Matrices

Matrix A matrix is a rectangular array of elements. An array is a systematic arrangement of numbers or symbols in rows and columns. Matrices (the plural of matrix) may be used to display information and to solve systems of linear equations. The numbers in the rows and columns of a matrix are called the elements of the matrix.

Dimensions of a Matrix The dimensions of a matrix may be indicated with the notation r  s, where r is the number of rows and s is the number of columns of a matrix. A matrix that contains the same number of rows and columns is called a square matrix. Example: 3  3 square matrices:

Addition and Subtraction of Matrices Two matrices can only be added or subtracted if they have the same dimensions. The corresponding elements of the two matrices are either added or subtracted.

Example: Adding Matrices Example: Find A + B. Solution: A + B

Practice Problem: Determine A - B

Scalar Multiplication A matrix may be multiplied by a real number, a scalar, by multiplying each entry in the matrix by the real number. Determine 3A

Multiplication of Matrices Multiplication of matrices is possible only when the number of columns in the first matrix is the same as the number of rows of the second matrix. In general,

Example: Multiplying Matrices Solution:

Practice: Determine A X B

Example: Identity Matrix in Multiplication Use the multiplicative identity matrix for a 2  2 matrix and matrix A to show that Solution: The identity matrix is

Example: Identity Matrix in Multiplication continued

Multiplicative Identity Matrix Square matrices have a multiplicative identity matrix. The following are the multiplicative identity for a 2 by 2 and a 3 by 3 matrix. For any square matrix A, A  I = I  A = A.

Homework P. 411 # 12 – 54 (x3)