Let A and B be symmetric nxn matrices. For each of the following, determine whether the given matrix must be symmetric or could be nonsymmetric. a) C = A + Bb) D = A^2c) E = ABd) F = ABAe) G = AB + BAf) H = AB-BA

All the matrices are symmetric.

Step-by-step explanation:

Symmetric matrices are those matrices that have equal dimensions, i.e. the number of rows is same as the number of columns. They are also known as square matrices.

It is provided that A and B are symmetric n × n matrices.

(i) To multiply two matrices of any order, the number of rows of the first matrix must be same as the number of columns of the second matrix.

Suppose X is a 2 × 3 matrix and Y is a 3 × 2.

Then:

Similarly the product AB will be a n × n matrix.

(ii) To add two matrices the dimension of both the matrix must be same.

Similarly the sum of matrix A and B will be a n × n matrix.

(a)

C = A + B

As shown above in point (ii), the sum of matrix A and B will be a n × n matrix.

So, matrix C is a n × n matrix.

Thus, the matrix C is symmetric.

(b)

D = A² = A × A

A is a n × n matrix.

On multiplying A by A, we will multiplying two square matrix of order n.

So, matrix D will also be a n × n matrix.

Thus, the matrix D is symmetric.

(c)

E = AB

As shown above in point (i), the product of matrix A and B will be a n × n matrix.

So, matrix E is a n × n matrix.

Thus, the matrix E is symmetric.

(d)

F = ABA = (AB) × A

As shown above in point (i), the product AB will be a n × n matrix.

The next step would be to multiply AB and A.

Both are n × n matrices.

So, their product will also be a n × n matrix.

Thus, the matrix F is symmetric.

(e)

G = AB + BA

As shown above in point (i), the product AB will be a n × n matrix.

Similarly, the product BA will be a n × n matrix.

Then the sum of two n × n matrices will also be a n × n matrix.

Thus, the matrix G is symmetric.

(f)

H = AB - BA

As shown above in point (i), the product AB will be a n × n matrix.

Similarly, the product BA will be a n × n matrix.

Now, as shown above in point (ii), the sum of matrix A and B will be a n × n matrix.

Similarly, the difference of matrix A and B will also be a n × n matrix.

Then the difference of matrices AB and BA will also be a n × n matrix.

Thus, the matrix H is symmetric.

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