Chapter 3 · Matrices

Exercise 3.3 · Q7: Show one matrix is symmetric and another skew symmetric

Exercise 3.3Symmetric and skew symmetric matrices · Section 3.6

The question

(i) Show that the matrix

A = \begin{bmatrix} 1 & -1 & 5 \\ -1 & 2 & 1 \\ 5 & 1 & 3 \end{bmatrix}

is a symmetric matrix.

(ii) Show that the matrix

A = \begin{bmatrix} 0 & 1 & -1 \\ -1 & 0 & 1 \\ 1 & -1 & 0 \end{bmatrix}

is a skew symmetric matrix.

Exam answer

(i)

A' = \begin{bmatrix} 1 & -1 & 5 \\ -1 & 2 & 1 \\ 5 & 1 & 3 \end{bmatrix} = A

$\therefore$ $A' = A$ ⁠, so $A$ is a symmetric matrix.

(ii)

\begin{aligned} A' &= \begin{bmatrix} 0 & -1 & 1 \\ 1 & 0 & -1 \\ -1 & 1 & 0 \end{bmatrix} \\ &= -\begin{bmatrix} 0 & 1 & -1 \\ -1 & 0 & 1 \\ 1 & -1 & 0 \end{bmatrix} \\ &= -A \end{aligned}

$\therefore$ $A' = -A$ ⁠, so $A$ is a skew symmetric matrix.

Want the reason behind every step, the basics and the checks?

The one idea this problem is built on

Picture the main diagonal as a mirror. Transposing swaps each entry with its mirror image across that diagonal. A matrix is symmetric when every entry equals its mirror image ( $a_{ji} = a_{ij}$ ⁠), so transposing changes nothing: $A' = A$ ⁠. It is skew symmetric when every entry is the negative of its mirror image ( $a_{ji} = -a_{ij}$ ⁠), so transposing flips every sign: $A' = -A$ ⁠. In that case each diagonal entry must equal its own negative, so the diagonal is all zeros.

Given, to find, and the plan

Given: Two $3\times3$ (square) matrices.
To show: (i) $A' = A$ ⁠; (ii) $A' = -A$ ⁠.
Plan: For each: (1) write $A'$ by turning rows into columns; (2) compare it with $A$ (or with $-A$ ⁠) entry by entry; (3) state the conclusion using the definition.

Basics you need

1The transpose

$(A')_{ij} = a_{ji}$ ⁠: rows become columns, and each entry moves to its mirror position across the main diagonal. Basics: mirror patterns

2Symmetric and skew symmetric

A square matrix is symmetric if $A' = A$ ⁠, that is $a_{ij} = a_{ji}$ for all $i, j$ ⁠. It is skew symmetric if $A' = -A$ ⁠, that is $a_{ji} = -a_{ij}$ for all $i, j$ ⁠; then $a_{ii} = -a_{ii}$ ⁠, so $a_{ii} = 0$ ⁠. Lesson: symmetric and skew symmetric

3The negative of a matrix

$-A = (-1)A$ ⁠: change the sign of every entry. Basics: scalar multiples

The dashed line is the main diagonal. Entries of the same colour are mirror images,

a_{ij}

and

a_{ji}

⁠. In (i) each pair is equal; in (ii) each pair has opposite signs and the diagonal entries are 0.

Part (i): the symmetric matrix

(i) Write $A'$

$A$ is $3\times3$ ⁠, so it is square and the question of symmetry makes sense. Its rows are $1, -1, 5$ ⁠; $-1, 2, 1$ ⁠; $5, 1, 3$ ⁠. Writing them as columns:
$A' = \begin{bmatrix} 1 & -1 & 5 \\ -1 & 2 & 1 \\ 5 & 1 & 3 \end{bmatrix}.$
Symmetric and skew symmetric are defined only for square matrices, because $A'$ must have the same order as $A$ for $A' = A$ to be possible.

Transpose, Definition 3 (Basic 1). Lesson: transpose
(i) Compare $A'$ with $A$ ⁠, pair by pair

The diagonal entries 1, 2, 3 are unchanged by transposing. The three mirror pairs are
- $a_{12} = -1$ and $a_{21} = -1$ ⁠: equal;
- $a_{13} = 5$ and $a_{31} = 5$ ⁠: equal;
- $a_{23} = 1$ and $a_{32} = 1$ ⁠: equal.
So every entry of $A'$ equals the corresponding entry of $A$ ⁠, and $A' = A$ ⁠.

A $3\times3$ matrix has 3 diagonal entries and 3 mirror pairs off the diagonal. Checking all three pairs covers every position, which is what "for all $i, j$ " requires here.
(i) Conclude

$A$ is square and $A' = A$ ⁠, so by Definition 4 $A$ is a symmetric matrix.

Definition 4, Section 3.6 (Basic 2). Lesson: symmetric matrices

Part (ii): the skew symmetric matrix

(ii) Write $A'$ and $-A$

The rows of $A$ ⁠, $0, 1, -1$ ⁠; $-1, 0, 1$ ⁠; $1, -1, 0$ ⁠, become columns:
$A' = \begin{bmatrix} 0 & -1 & 1 \\ 1 & 0 & -1 \\ -1 & 1 & 0 \end{bmatrix}.$
Changing the sign of every entry of $A$ ⁠:
$-A = \begin{bmatrix} 0 & -1 & 1 \\ 1 & 0 & -1 \\ -1 & 1 & 0 \end{bmatrix}.$
To test $A' = -A$ we need both matrices written out. Note that $-0 = 0$ ⁠, so the zero diagonal stays zero.

$-A = (-1)A$ ⁠, Section 3.4.2 (Basic 3).
(ii) Compare, pair by pair
- Diagonal: $a_{11} = a_{22} = a_{33} = 0$ ⁠, as a skew symmetric matrix requires.
- $a_{12} = 1$ and $a_{21} = -1$ ⁠: opposite.
- $a_{13} = -1$ and $a_{31} = 1$ ⁠: opposite.
- $a_{23} = 1$ and $a_{32} = -1$ ⁠: opposite.
Hence every entry of $A'$ equals the corresponding entry of $-A$ ⁠: $A' = -A$ ⁠.

The condition $a_{ji} = -a_{ij}$ must hold for every $i$ and $j$ ⁠, including $i = j$ ⁠, where it forces $a_{ii} = 0$ ⁠. Here all three diagonal entries are 0 and all three off-diagonal pairs are opposite, so every case is covered.
(ii) Conclude

$A$ is square and $A' = -A$ ⁠, so by Definition 5 $A$ is a skew symmetric matrix. (It is not symmetric, since $A' \ne A$ ⁠: for example $a_{12} = 1$ but $a_{21} = -1$ ⁠.)

Definition 5, Section 3.6. Lesson: skew symmetric matrices

Result

(i) $A' = A$ ⁠, so $A$ is symmetric. (ii) $A' = -A$ ⁠, so $A$ is skew symmetric.

Check it with numbers

Add a matrix to its transpose. In (i), $A + A' = 2A$ ⁠, whose (1,3) entry is $5 + 5 = 10 = 2\times5$ ⁠. ✓ In (ii), $A + A'$ should be the zero matrix: (1,2) entry $1 + (-1) = 0$ ⁠, (1,3) entry $-1 + 1 = 0$ ⁠, (2,3) entry $1 + (-1) = 0$ ⁠, diagonal $0 + 0 = 0$ ⁠. ✓

Common mistakes

Checking only one pair. All three off-diagonal pairs (and, for skew symmetric, the diagonal) must be checked.
Thinking a skew symmetric matrix can have a nonzero diagonal. $a_{ii} = -a_{ii}$ forces $a_{ii} = 0$ ⁠.
Comparing rows with rows. Symmetry compares row $i$ with column $i$ ⁠: row 1 of (i), $1, -1, 5$ ⁠, equals column 1, $1, -1, 5$ ⁠.

What to take away

Symmetric: $A' = A$ ⁠, mirror pairs equal. Skew symmetric: $A' = -A$ ⁠, mirror pairs opposite, diagonal zero.
For a $3\times3$ matrix there are exactly three mirror pairs to check.
Row $i$ of a symmetric matrix reads the same as column $i$ ⁠.

Every fact used, and where it comes from

Fact	Used in	Source
Transpose $(A')_{ij} = a_{ji}$	Part (i), Step 1; Part (ii), Step 1	Definition 3, Section 3.5 (Lesson: transpose)
Symmetric: square and $A' = A$	Part (i), Step 3	Definition 4, Section 3.6 (Lesson)
Skew symmetric: square and $A' = -A$ ⁠; diagonal entries zero	Part (ii), Steps 2–3	Definition 5, Section 3.6
Negative of a matrix $-A = (-1)A$	Part (ii), Step 1	Section 3.4.2

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Exercise 3.3 · Q7: Show one matrix is symmetric and another skew symmetric

Basics you need

Part (i): the symmetric matrix

(i) Write A′A'A′

(i) Compare A′A'A′ with AAA⁠, pair by pair

(i) Conclude

Part (ii): the skew symmetric matrix

(ii) Write A′A'A′ and −A-A−A

(ii) Compare, pair by pair

(ii) Conclude

Every fact used, and where it comes from

(i) Write $A'$

(i) Compare $A'$ with $A$ ⁠, pair by pair

(ii) Write $A'$ and $-A$