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Matrix Arithmetic Help

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By McGraw-Hill Professional
Updated on Oct 4, 2011

Introduction to Matrix Arithmetic

A matrix is an array of numbers or symbols made up of rows and columns. Matrices are used in science and business to represent several variables and relationships at once. For example, suppose there are three brands of fertilizers that provide different levels of three minerals that a gardener might need. The following matrix shows how much of each mineral is provided by each brand.

 Mineral A Mineral B Mineral C Brand X 6 2 1 Brand Y 2 1 2 Brand Z 1 3 6

We will learn some matrix arithmetic as well as two matrix methods used to solve systems of linear equations. Most of the calculations are tedious. Fortunately graphing calculators and computer programs (including spreadsheets) can do most of them.

Matrix Arithmetic

The numbers in a matrix are called cells or entries. A matrix’s size is given by the number of rows and columns it has. For example, a matrix that has two rows and three columns is called a 2 × 3 (pronounced “2 by 3”) matrix. A matrix that has the same number of rows as columns is called a square matrix.

Two matrices need to be the same size before we can add them or find their difference. The sum of two or more matrices is the sum of their corresponding entries.

Subtract one matrix from another by subtracting their corresponding entries.

The scalar product of a matrix is a matrix whose entries are multiplied by a fixed number.

Matrix Multiplication

It might seem that matrix multiplication is carried out the same way addition and subtraction are—multiply their corresponding entries. This operation is not very useful. The matrix multiplication operation that is useful requires more work. Two matrices do not need to be the same size, but the number of columns of the first matrix must be the same as the number of rows of the second matrix. This is because we get the entries of the product matrix by multiplying the rows of the first matrix by the columns of the second matrix. Here, we will multiply a 3 × 3 matrix by a 3 × 2 matrix.

Row 1 of the first matrix is A B C and Column 1 of the second matrix . The first entry on the product matrix is Row 1 × Column 1, which is this sum.

Examples

• An identity matrix is a square matrix with 1s on the main diagonal (from the upper left corner to the bottom right corner) and 0s everywhere else. The following are the 2 × 2 and 3×3 identity matrices.

If we multiply any matrix by its corresponding identity matrix, we will get the original matrix back.

Matrix multiplication is not commutative. Reversing the order of the multiplication usually gets a different matrix, if the multiplication is even possible.

The matrix

is not the same as

Practice

Compute the following.

Solutions

Practice problems for this concept can be found at: Matrices Practice Test.

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