3A.4 Matrix Inverse

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An operation similar to scalar division can be defined in terms of the inverse of a matrix. A proper inverse

is defined only for a square matrix and, even for a square matrix, an inverse may not exist. The inverse of

a matrix is defined as follows.

Suppose that a square matrix A has the inverse B. Then, these must satisfy the equation

AB ¼ I ð3A:12Þ

or equivalently

BA ¼ I ð3A:13Þ

where I is the identity matrix, as defined before.

The inverse of A is denoted by A 21. The inverse exists for a matrix if and only if the determinant of

the matrix is nonzero. Such matrices are termed nonsingular. We shall discuss the determinant in

Section 3A.6. Before explaining a method for determining the inverse of a matrix, let us verify that

2 1

1 1

" #

is the inverse of

1 21

21 2

" #

To show this, we simply multiply the two matrices and show that the product is the second-order unity

matrix. Specifically,

1 21

21 2

" #

2 1

1 1

" #

¼

1 0

0 1

" #

or

2 1

1 1

" #

1 21

21 2

" #

¼

1 0

0 1

" #

3A.4.1 Matrix Transpose

The transpose of a matrix is obtained by simply interchanging the rows and the columns of the matrix.

The transpose of A is denoted by AT.

For example, the transpose of the 2 £ 3 matrix

A ¼

1 22 3

22 2 0

" #

is the 3 £ 2 matrix

AT ¼

1 22

22 2

3 0

2

664

3

775

3-46 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

Note that the first row of the original matrix has become the first column of the transposed matrix, and

the second row of the original matrix has become the second column of the transposed matrix.

If AT ¼ A; then we say that the matrix A is symmetric. Another useful result on the matrix transpose is

expressed by

ðABÞT ¼ BTAT ð3A:14Þ

It follows that the transpose of a matrix product is equal to the product of the transposed matrices,

taken in the reverse order.

3A.4.2 Trace of a Matrix

The trace of a square matrix is given by the sum of the diagonal elements. The trace of matrix A is

denoted by trðAÞ:

trðAÞ ¼

X

i

aii ð3A:15Þ

For example, the trace of the matrix

A ¼

22 3 0

4 24 1

21 0 3

2

664

3

775

is given by

trðAÞ ¼ ð22Þ þ ð24Þ þ 3 ¼ 23

3A.4.3 Determinant of a Matrix

The determinant is defined only for a square matrix. It is a scalar value computed from the elements of

the matrix. The determinant of a matrix A is denoted by detðAÞ or lAl:

Instead of giving a complex mathematical formula for the determinant of a general matrix in terms of

the elements of the matrix, we now explain a way to compute the determinant.

First, consider the 2 £ 2 matrix

A ¼

a11 a12

a21 a22

" #

Its determinant is given by

detðAÞ ¼ a11a22 2 a12a21

Next, consider the 3 £ 3 matrix

A ¼

a11 a12 a13

a21 a22 a23

a31 a32 a33

2

664

3

775

Its determinant can be expressed as

detðAÞ ¼ a11M11 2 a12M12 þ a13M13

where the minors of the associated matrix elements are defined as

M11 ¼ det

a22 a23

a32 a33

" #

; M12 ¼ det

a21 a22

a31 a32

" #

; M13 ¼ det

a21 a22

a31 a32

" #

Modal Analysis 3-47

© 2005 by Taylor & Francis Group, LLC

Note that Mij, the determinant of the matrix, is obtained by deleting the ith row and the jth column of

the original matrix. The quantity Mij is known as the minor of the element aij of the matrix A. If we attach

a proper sign to the minor depending on the position of the corresponding matrix element, we have a

quantity known as the cofactor. Specifically, the cofactor, Cij; corresponding to the minor, Mij; is given by

Cij ¼ ð21ÞiþjMij ð3A:16Þ

Hence, the determinant of the 3 £ 3 matrix may be given by

detðAÞ ¼ a11C11 þ a12C12 þ a13C13

Note that in the two formulas given above for computing the determinant of a 3 £ 3 matrix, we have

expanded along the first row of the matrix. We get the same answer, however, if we expand along any row

or any column. Specifically, when expanded along the ith row, we have

detðAÞ ¼ ai1Ci1 þ ai2Ci2 þ ai3Ci3

Similarly, if we expand along the jth column, we have

detðAÞ ¼ a1jC1j þ a2jC2j þ a3jC3j

These ideas of computing a determinant can be easily extended to 4 £ 4 and higher-order matrices in a

straightforward manner. Hence, we can write

detðAÞ ¼

X

j

aijCij ¼

X

i

aijCij ð3A:17Þ

3A.4.4 Adjoint of a Matrix

The adjoint of a matrix is the transpose of the matrix whose elements are the cofactors of the

corresponding elements of the original matrix. The adjoint of matrix A is denoted by adjðAÞ:

As an example, in the 3 £ 3 case, we have

adjðAÞ ¼

C11 C12 C13

C21 C22 C23

C31 C32 C33

2

664

3

775

T

¼

C11 C21 C31

C12 C22 C32

C13 C23 C33

2

664

3

775

In particular, it is easily seen that the adjoint of the matrix

A ¼

1 2 21

0 3 2

1 1 1

2

664 3 775

is given by

adjðAÞ ¼

1 2 23

23 2 1

7 22 3

2

664

3

775

T

Accordingly, we have

adjðAÞ ¼

1 23 7

2 2 22

23 1 3

2

664

3

775

3-48 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

Hence, in general

adjðAÞ ¼ ½Cij􀀉T ð3A:18Þ

3A.4.5 Inverse of a Matrix

At this juncture, it is appropriate to give a formula for the inverse of a square matrix. Specifically,

A21 ¼

adjðAÞ

detðAÞ ð3A:19Þ

Hence, in the 3 £ 3 matrix example given before, since we have already determined the adjoint, it

remains only to compute the determinant in order to obtain the inverse. Now, expanding along the first

row of the matrix, the determinant is given by

detðAÞ ¼ 1 £ 1 þ 2 £ 2 þ ð21Þ £ ð23Þ ¼ 8

Accordingly, the inverse is given by

A21 ¼

1

8

1 23 7

2 2 22

23 1 3

2

664

3

775

For two square matrices A and B we have

ðABÞ21 ¼ B21A21 ð3A:20Þ

As a final note, if the determinant of a matrix is zero, the matrix does not have an inverse. Then we say

that the matrix is singular. Some important matrix properties are summarized in Box 3A.1.

Box 3A.1

SUMMARY OF MATRIX PROPERTIES

Addition: Am£n þ Bm£n ¼ Cm£n

Multiplication: Am£nBn£r ¼ Cm£r

Identity: AI ¼ IA ¼ A ) I is the identity matrix

Note: AB ¼ 0 )⁄ A ¼ 0 or B ¼ 0 in general

Transposition: CT ¼ ðABÞT ¼ BTAT

Inverse: AP ¼ I ¼ PA ) A ¼ P21 and P ¼ A21

ðABÞ21 ¼ B21A21

Commutativity: AB – BA in general

Associativity: ðABÞC ¼ AðBCÞ

Distributivity: CðA þ BÞ ¼ CA þ CB

Distributivity: ðA þ BÞD ¼ AD þ BD

Modal Analysis 3-49

© 2005 by Taylor & Francis Group, LLC

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