Transpose Matrix

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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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Every one of us uses matrices nearly everyday in our lives and probably unaware of it. Matrices are commonly used in computers for their 3D graphics. Most of the matrices that are used are either 3×3 or 4×4 matrices and are computed by either rotation matrices or translation matrices. The matrices that are used are an array that holds numbers, commonly called a 3×3 array or 4×4 ...

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