Answer :

We need to find a matrix of order 2 × 2 which is both symmetric and skew-symmetric.

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}

Now, let us understand what skew-symmetric matrix is.

A skew-symmetric matrix is a square matrix whose transpose equals its negative, that, it satisfies the condition

A skew symmetric matrix ⬄ A^{T} = -A

And,

A square matrix is a matrix with the same number of rows and columns. An n-by-n matrix is known as a square matrix of order n.

Take a 2 × 2 null matrix.

Say,

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.

So,

Since, A = A^{T}.

∴, A is symmetric.

Take the same matrix and multiply it with -1.

Let us take transpose of the matrix –A.

So,

Since,

A^{T} = -A

∴, A is skew-symmetric.

Thus, A (a null matrix) of order 2 × 2 is both symmetric as well as skew-symmetric.

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