Basic Operations with Matrices
Properties of Matrices
• A matrix is a rectangular array of values consisting of intersecting rows and columns.
• An upper-case variable is used to represent a unique matrix much like a lower-case variable represents a unique value.
• The dimensions of a matrix are stated as the number of rows by the number of columns.

Example

• The name of the matrix in this example is Matrix X.
• The variable assigned to this matrix is arbitrary and is independent from that of the contained values.
• Matrix X is a 2 x 3 matrix.
• When verbalizing the dimensions of a matrix, read them like you would the dimensions of lumber or a room. This example would be read as “two by three”.
 In a 4 x 5 matrix, how many values are present? What is your answer?
• Only matrices with equal dimensions can be added.
• The addition of matrices is commutative. A + B = B + A
• The addition of matrices is associative. (A + B) + C = A + (B + C)

Examples
We will use the following three matrices do complete the example problems.

Subtraction:
• Only matrices with equal dimensions can be subtracted.
• Subtraction of matrices is not commutative.
• If one looks at subtraction as the addition of a negative, then the equation is commutative.
• Subtraction of matrices is not associative.
• If one looks at subtraction as the addition of a negative, then the equation is associative.

Examples
We will use the following three matrices do complete the example problems.

Scalar Multiplication:
• A number that is multiplying the matrix is called a Scalar
• The multiplication of a matrix and a scalar is commutative.
• The multiplication of a matrix and a scalar is associative.
• If dividing by a value, multiply by the value's inverse.

Examples
We will use the following three matrices do complete the example problems.