Matrices and Matrix Operations Math149 Skills Objectives, Lecture notes of Engineering Mathematics

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MATRICES AND MATRIX

OPERATIONS

MATH

SKILLS OBJECTIVES

At the end of this lesson, you are expected to demonstrate the following:

1. Perform the arithmetic operations of matrix addition, subtraction, scalar multiplication and multiplication.
2. Compute the transpose of the matrix.
3. Compute the trace of the matrix.
4. Determine whether the product of two given matrices is defined.
5. Compute matrix products using row-column method, the column method and the row method.
6. Express the product of a matrix and a column vector as a linear combination of the columns of the matrix.
7. Express a linear system as a matrix equation and identify the coefficient matrix.

Matrix Definition For example, a 3x4 matrix has entries written as A=

Transpose of a matrix If 𝐴 is any 𝑚 × 𝑛 matrix, then the transpose of 𝑨, denoted by 𝐴 𝑇 , is defined to be the 𝑛 × 𝑚 matrix that results from interchanging the rows and columns of 𝐴; that is, the first column of 𝐴 𝑇 is the first row of 𝐴, the second column of 𝐴 𝑇 is the second row of 𝐴, and so forth.

• Examples:

• Symmetric Matrix A special kind of square matrix that is equal to its transpose. (𝐴 = 𝐴 𝑇 ) Example: A=

𝑇 = =

• If A is a square matrix, then the trace of A , denoted by tr(A) is defined to be the sum of the entries on the main diagonal of A. The trace of A is undefined if A is not a square matrix.
• Example. Find the trace of A in the previous example. tr(A) = 1 + 4 + 6= 11

Matrix Addition and Subtraction

If A and B are matrices of the same size , then the sum A+B is the matrix obtained by adding the entries of B to the corresponding entries of A, and the difference A-B i s the matrix obtained by subtracting the entries of B from the corresponding entries of A. Matrices of different sizes cannot be added or subtracted. In matrix notation, If A = 𝑎𝑖𝑗 and B= 𝑏𝑖𝑗 have the same size, then 𝐴 + 𝐵 (^) 𝑖𝑗 = 𝐴 (^) 𝑖𝑗 + 𝐵 (^) 𝑖𝑗 = 𝑎𝑖𝑗 + 𝑏𝑖𝑗 and 𝐴 − 𝐵 (^) 𝑖𝑗 = 𝐴 (^) 𝑖𝑗 − 𝐵 (^) 𝑖𝑗 = 𝑎𝑖𝑗 − 𝑏𝑖𝑗 Ex: Let A = 1 2 3 4 and B = 5 6 7 8 then A-B = 1 − 5 2 − 6 3 − 7 4 − 8 = − 4 − 4 − 4 − 4

Scalar Multiplication

If A is any matrix and c is any scalar, then the product cA is the matrix obtained by multiplying each entry of the matrix A by c. The matrix cA is said to be a scalar multiple of A. If A = 𝑎𝑖𝑗 , then 𝑐𝐴 (^) 𝑖𝑗= c 𝐴 (^) 𝑖𝑗 =c𝑎𝑖𝑗. Let A=

then 3A =

Example: no.3 p. Consider the matrices A =

, B =

, C =

D =

E =

In each part, compute the given expression (where possible). (a) D+E (b) D-E (c) 5A (d) - 9D (e) 2B-C (f) 7E-3D (g) 2(D+5E) (h) 2𝐴 𝑇 +C (i) (𝐷 − 𝐸) 𝑇 (j)(2𝐸 𝑇

• 3 𝐷 𝑇 ) (k) tr(D) (l) tr(D-E) (m) 2tr(4B)

Matrix Multiplication The definition of matrix multiplication requires that the number of columns of the first factor 𝐴 be the same as the number of rows of the second factor 𝐵 in order to form the product 𝐴𝐵. If this condition is not satisfied, the product is undefined. A convenient way to determine whether a product of two matrices is defined is to write down the size of the first factor and, to the right of it, write down the size of the second factor. If the inside numbers are the same, then the product is defined. The outside numbers then give the size of the product.

Matrix Multiplication

• If 𝐴 is an 𝑚 × 𝑟 matrix and 𝐵 is an 𝑟 × 𝑛 matrix, then the product 𝐴𝐵 is the 𝑚 × 𝑛 matrix whose entries are determined as follows. To find the entry in row 𝑖 and column 𝑗 of 𝐴𝐵, single out row 𝑖 from the matrix 𝐴 and column 𝑗 from the matrix 𝐵. Multiplying the corresponding entries from the row and column together, and then add up the resulting products. 𝑨𝑩 (^) 𝒊𝒋 = 𝒂𝒊𝟏𝒃𝒊𝒋 + 𝒂𝒊𝟐𝒃𝟐𝒋 + 𝒂𝒊𝟑𝒃𝟑𝒋 + … + 𝒂𝒊𝒓𝒃𝒓𝒋

Example

• The entry in row 1 and column 4 of 𝐴𝐵 is computed as follows:
• The computations for the remaining entries are

• Matrix Products as Linear Combinations
• If 𝐴 1 , 𝐴, 2 …,𝐴𝑛 are matrices of the same size, and if 𝑐 1 , 𝑐 2 ,…,𝑐 3 are scalars, then the expression of the form 𝑐 1 𝐴 1 + 𝑐 2 𝐴 2 + ⋯ + 𝑐𝑟𝐴𝑟 is called a linear combination of 𝐴 1 , 𝐴, 2 …,𝐴𝑛 , with coefficients 𝑐 1 , 𝑐 2 ,…,𝑐 3..

Theorem: If A is an mxn matrix, and if x is an nx1 column vector, then the product Ax can be expressed as a linear combination of the column vectors of A in which the coefficients are the entries of x. Example: Matrix Product as Linear Combinations

• The matrix product − 1 3 2 1 2 − 3 2 1 − 2

can be written as the following linear combination of column vectors 2