Answer :

We are given that,

is symmetric matrix.

We need to find the values of a and b.

We must understand what symmetric matrix is.

A symmetric matrix is a square matrix that is equal to its transpose.

A symmetric matrix ⬄ A = A^{T}

This means, we need to find the transpose of matrix A.

Let us take transpose of the matrix A.

We know that, the transpose of a matrix is a new matrix whose rows are the columns of the original.

We have,

1^{st} row of matrix A = (0 2b -2)

2^{nd} row of matrix A = (3 1 3)

3^{rd} row of matrix A = (3a 3 -1)

For matrix A^{T}, it will become

1^{st} column of A^{T} = 1^{st} row of A = (0 2b -2)

2^{nd} column of A^{T} = 2^{nd} row of A = (3 1 3)

3^{rd} column of A^{T} = 3^{rd} row of A = (3a 3 -1)

Now, as A = A^{T}.

Substituting the matrices A and A^{T}, we get

We know by the property of matrices,

This implies,

a_{11} = b_{11}, a_{12} = b_{12}, a_{21} = b_{21} and a_{22} = b_{22}

Applying this property, we can write

2b = 3 …(i)

-2 = 3a …(ii)

3 = 2b

3a = -2

We can find a and b from equations (i) and (ii).

From equation (i),

2b = 3

From equation (ii),

-2 = 3a

Thus, we get and .

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