Q.(i) Show that the matrix $A = \begin{bmatrix} 1 & -1 & 5 \\ -1 & 2 & 1 \\ 5 & 1 & 3 \end{bmatrix}$ is a symmetric matrix.

Concept: Symmetric and Skew Symmetric Matrices

A matrix $A$ is symmetric if $A^T = A$, and skew symmetric if $A^T = -A$.

(i) For $A = \begin{bmatrix} 1 & -1 & 5 \\ -1 & 2 & 1 \\ 5 & 1 & 3 \end{bmatrix}$, compute $A^T$ by swapping rows and columns:

$A^T = \begin{bmatrix} 1 & -1 & 5 \\ -1 & 2 & 1 \\ 5 & 1 & 3 \end{bmatrix}$.

Since $A^T = A$, the matrix is symmetric.

(ii) For $A = \begin{bmatrix} 0 & 1 & -1 \\ -1 & 0 & 1 \\ 1 & -1 & 0 \end{bmatrix}$, compute $A^T = \begin{bmatrix} 0 & -1 & 1 \\ 1 & 0 & -1 \\ -1 & 1 & 0 \end{bmatrix}$. …

The idea is straightforward: transpose the matrix and compare it with the original. For a symmetric matrix, the entry at $(i,j)$ must equal the entry at $(j,i)$ — the matrix is mirrored across the main diagonal. For a skew symmetric matrix, the entry at $(i,j)$ must be the negative of the entry at $(j,i)$, and the diagonal entries must all be zero.

Let's check each case.

Part (i): $A = \begin{bmatrix} 1 & -1 & 5 \\ -1 & 2 & 1 \\ 5 & 1 & 3 \end{bmatrix}$

1. Write down the transpose. The transpose swaps rows and columns: the first row becomes the first column, the second row becomes the second column, and so on.

2. Compare $A^T$ with $A$. Look at each position:

   • $(1,1)$: $1 = 1$
   • $(1,2)$: $-1 = -1$
   • $(1,3)$: $5 = 5$
   • $(2,1)$: $-1 = -1$
   • $(2,2)$: $2 = 2$
   • $(2,3)$: $1 = 1$
   • $(3,1)$: $5 = 5$
   • $(3,2)$: $1 = 1$
   • $(3,3)$: $3 = 3$

   Every entry matches exactly. So $A^T = A$.

Part (ii): $A = \begin{bmatrix} 0 & 1 & -1 \\ -1 & 0 & 1 \\ 1 & -1 & 0 \end{bmatrix}$

1. Write the transpose:

2. Now check if $A^T = -A$. First compute $-A$:

3. Compare $A^T$ and $-A$ entry by entry:
   • $(1,1)$: $0 = 0$
   • $(1,2)$: $-1 = -1$
   • $(1,3)$: $1 = 1$
   • $(2,1)$: $1 = 1$ …

Method: Testing whether a matrix is symmetric or skew symmetric

Compute the transpose $A'$ and compare it with $A$: if $A'=A$ the matrix is symmetric; if $A'=-A$ (which forces zeros on the diagonal) it is skew symmetric.

Steps

Step 1: Form the transpose

Interchange rows and columns to write $A'$.

Step 2: Compare with $A$ and with $-A$ …

Common Mistakes

Mistake 1: Comparing $A'$ with $A$ when testing for skew symmetry

Why it's wrong: skew symmetry requires $A'=-A$, not $A'=A$; you must compare against the negative of the matrix. Correct approach: build $-A$ and check $A'=-A$ entry-by-entry.

Mistake 2: Ignoring the zero-diagonal requirement for skew symmetry …