Mathematics : Applications of Matrices and Determinants | Inverse of a Non-Singular Square Matrix: Solved Example Problems

If A = , verify that A (adj A) = (adj A) A = | A | I3 .

Solution

Example 1.2

If A = is non-singular, find Aâˆ’1

Solution

We first find adj *A*. By definition, we get

Find the inverse of the matrix

Solution

Example 1.4

If *A *is a non-singular matrix of odd order, prove that |adj A| is positive.

Solution

Let A be a non-singular matrix of order 2*m* + 1 , where *m* = 0,1, 2, .. . . Then, we get |A| â‰ 0 and, by theorem 1.9 (ii), we have |adj A| = |A|(2m+1)-1 = |A|2m

Since |A|2m is always positive, we get that |adj A| is positive.

Example 1.5

Find a matrix *A *if adj( *A*) =

Solution

Example 1.6

If adj *A *= find *A*âˆ’1.

Solution

Example1.7

If *A *is symmetric, prove that adj *A *is also symmetric.

Solution

Suppose A is symmetric. Then, AT = A and so, by theorem 1.9 (vi), we get

adj (AT) = (adj A) T â‡’ adj A = (adj A)T â‡’ adj A is symmetric

Verify the property ( *AT *)âˆ’1 = ( *A*âˆ’1 )*T** *with *A *= .

For the given A, we get |A |= (2) (7) - (9)(1) = 14 âˆ’ 9 = 5 .

From (1) and (2), we get (A-1) = (AT)-1. Thus, we have verified the given property.

Example 1.9

Verify ( *AB*)âˆ’1 = *B*âˆ’1 *A*âˆ’1 with

Solution

As the matrices in (1) and (2) are same, (AB) âˆ’1 = B-1 A-1 is verified.

If *A = **, *find *x *and *y *such that *A*2 + *xA *+ *yI2=O2,* Hence, find *A*âˆ’1.

Solution

So, we get 22 + 4x + y =0, 31+5x+y=0, 27+3x=0 and 18+2x=0

Hence x = âˆ’9 and y =14.Then, we get A2 - 9A + 14I2 = O2

Post-multiplying this equation by Aâˆ’1 , we get A â€“ 9I2 + 14A-1 = O2. Hence, we get

Prove that is orthogonal.

Similarly, we get ATA = I2 . Hence AAT = ATA = I2 â‡’ A is orthogonal.

If A = , is orthogonal, find *a*, *b *and *c *, and hence *A*âˆ’1.

If A is orthogonal, then AAT = AT A = I3 . So, we have

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12th Mathematics : UNIT 1 : Applications of Matrices and Determinants : Inverse of a Non-Singular Square Matrix: Solved Example Problems |

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