Transpose Matrix

The transpose of one matrix is another matrix that is obtained by using by using rows from the first matrix as columns in the second matrix.

For example,

A = 111 222

333 444

555 666

A’ = 111 333 555

222 444 666

Note that the order of a matrix is reversed after it has been transposed. Matrix A is a 3 x 2 matrix, but matrix A’ is a 2 x 3 matrix.

Vectors

Vectors are a type of matrix having only one column or one row.

. A matrix with one column (an m × 1 matrix) is called a column vector and one row (a 1 × n matrix) is called a row vector.For example, Matrix a is a column vector, and matrix a’ is a row vector.

a = 11

12

33

a’ = 11 22 33

Square Matrices

A square matrix is an n x n matrix; that is, a matrix with the same number of rows as columns. In this section, we describe several special kinds of square matrix.

• Symmetric matrix. If the transpose of a matrix is equal to itself, that matrix is said to be symmetric. Two examples of symmetric matrices appear below.

A = A’ = 1 2

2 3

B = B’ = 5 6 7

6 3 2

7 2 1

Diagonal matrix. A diagonal matrix is a special kind of symmetric matrix. It is a symmetric matrix with zeros in the off-diagonal elements. Two diagonal matrices are shown below.

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A = 1 0

0 3

B = 5 0 0

0 3 0

0 0 1

Scalar matrix. A scalar matrix is a special kind of diagonal matrix. It is a diagonal matrix with equal-valued elements along the diagonal. Two examples of a scalar matrix appear below.

A = 3 0

0 3

B = 5 0 0

0 5 0

0 0 5

Matrix Dimension

The number of rows and columns that a matrix has is called its dimension or its order. By convention, rows are listed first; and columns, second. Thus, we would say that the dimension (or order) of the matrix below is 3 x 4, meaning that it has 3 rows and 4 columns.

21 62 33 93

44 95 66 13

77 38 79 33

Matrix Addition and Matrix Subtraction

How to Add and Subtract Matrices

Two matrices may be added or subtracted only if they have the same dimension; that is, they must have the same number of rows and columns.

. For example, consider matrix A and matrix B.

A = 1 2 3

7 8 9

B = 5 6 7

3 4 5

Both matrices have the same number of rows and columns (2 rows and 3 columns), so they can be added and subtracted. Thus,

A + B = 1 + 5 2 + 6 3 + 7

7 + 3 8 + 4 9 + 5

= 6 8 10

10 12 14

And,

A – B = 1 – 5 2 – 6 3 – 7

7 – 3 8 – 4 9 – 5

= -4 -4 -4

4 4 4

Matrix Multiplication

In matrix algebra, there are two kinds of matrix multiplication: multiplication of a matrix by a number or scalar and multiplication of a matrix by another matrix.

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For example, if x is 5, and the matrix A is:

A = 100 200

300 400

Then,

xA = 5A = 5 100 200

300 400

= 5 * 100 5 * 200

5 * 300 5 * 400

= 500 1000

1500 2000

= B

How to Multiply a Matrix by a Matrix

The matrix product AB is defined only when the number of columns in A is equal to the number of rows in B. Similarly, the matrix product BA is defined only when the number of columns in B is equal to the number of rows in A.Ex-

A = 0 1 2

3 4 5

B = 6 7

8 9

10 11

AB = C = 28 31

100 112

Identity Matrix

The identity matrix is an n x n diagonal matrix with 1’s in the diagonal and zeros everywhere else. The identity matrix is denoted by I or In. Two identity matrices appear.

I2 = 1 0

0 1

I3 = 1 0 0

0 1 0

0 0 1

The identity matrix has a unique talent. Any matrix that can be premultiplied or postmultiplied by I remains the same; that is:

AI = IA = A