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 = AT


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,


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


2nd row of matrix A = (3 1 3)


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


For matrix AT, it will become


1st column of AT = 1st row of A = (0 2b -2)


2nd column of AT = 2nd row of A = (3 1 3)


3rd column of AT = 3rd row of A = (3a 3 -1)



Now, as A = AT.


Substituting the matrices A and AT, we get



We know by the property of matrices,



This implies,


a11 = b11, a12 = b12, a21 = b21 and a22 = b22


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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