NCERT Class 12-Mathematics: Exemplar Chapter –3 Matrices Part 2

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3.1.9 Symmetric Matrix and Skew Symmetric Matrix

(i) A square matrix is said to be symmetric if , that is, for all possible values of and .

(ii) A square matrix is said to be skew symmetric matrix if , that is for all possible values of and .

Note: Diagonal elements of a skew symmetric matrix are zero.

(iii) Theorem 1: For any square matrix A with real number entries, is a symmetric matrix and is a skew symmetric matrix.

(iv) Theorem 2: Any square matrix A can be expressed as the sum of a symmetric matrix and a skew symmetric matrix, that is

3.1.10 Invertible Matrices

(i) If A is a square matrix of order , and if there exists another square matrix B of the same order , such that , then, A is said to be invertible matrix and B is called the inverse matrix of A and it is denoted by .

Note :

1. A rectangular matrix does not possess its inverse, since for the products BA and AB to be defined and to be equal, it is necessary that matrices A and B should be square matrices of the same order.

2. If B is the inverse of A, then A is also the inverse of B.

(ii) Theorem 3: (Uniqueness of inverse) Inverse of a square matrix, if it exists, is unique.

(iii) Theorem 4: If A and B are invertible matrices of same order, then

3.1.11 Inverse of a Matrix using Elementary Row or Column Operations

To find using elementary row operations, write and apply a sequence of row operations on till we get,. The matrix B will be the inverse of A. Similarly, if we wish to find using column operations, then, write and apply a sequence of column operations on till we get, .

Note : In case, after applying one or more elementary row (or column) operations on , if we obtain all zeros in one or more rows of the matrix A on L.H.S., then does not exist.

3.2 Solved Examples

Short Answer (S.A)

Question1:

Construct a matrix whose elements are given

Answer:

Question 2:

If then which of the sums is defined?

Answer:

Only is defined since matrices of the same order can only be added.

Question 3:

Show that a matrix which is both symmetric and skew symmetric is a zero matrix.

Answer:

Let be a matrix which is both symmetric and skew symmetric.

Since A is a skew symmetric matrix, so .

Thus for all and , we have

Again, since A is a symmetric matrix, so .

Thus, for all and , we have

Therefore, from and , we get

for all and

i.e., for all i and j. Hence A is a zero matrix.

Question 4:

If , Find the value of

Answer:

We have

Question 5:

If A is invertible matrix, then show that for any scalar (non-zero), is invertible and

Answer:

We have

Hence (kA) is inverse of