LINEAR-ALGEBRA · Unit 5 · Video 2 · Interactive Practice

Matrix Transposes and Symmetric Matrices

IKey Formulas

Formula	Name	When to use
$(A^T)_{ij} = A_{ji}$	Transpose definition	Swap row and column indices
$(AB)^T = B^T A^T$	Reversal rule	Transpose of a product reverses order
$(A^T)^T = A$	Double transpose	Transposing twice returns the original
$A^T = A$	Symmetry test	A matrix that equals its own transpose

IIVisualization 1 — Rows Become Columns

Transposing sends each entry $A_{ij}$ to position $(j, i)$ — every row of $A$ turns into a column of $A^T$ .

a₁₁ 1 a₁₂ 3 a₁₃ 2 a₂₁ 4 a₂₂ 0 a₂₃ -1

IIIVisualization 2 — $R^T R$ Is Always Symmetric

Whatever numbers you put in $R$ , the product $R^T R$ comes out symmetric: its mirror entries always match.

r₁₁ 1 r₁₂ 3 r₂₁ 2 r₂₂ 3 r₃₁ 4 r₃₂ 1

💡 Why it must hold: $(R^T R)^T = R^T (R^T)^T = R^T R$ , so the product equals its own transpose.

IVVisualization 3 — Same Symmetry, Different Sizes

Both $R^T R$ and $R R^T$ are symmetric, but their sizes are set by the columns and rows of $R$ .

Matrix R

💡 The diagonal entries are sums of squares, so they are always non-negative.

VQuiz Questions

Question 1

Given $R = \begin{bmatrix} 1 & 3 \\ 2 & 3 \\ 4 & 1 \end{bmatrix}$ , what is entry $(1, 3)$ of $R^T$ ?

Recall: $(A^T)_{ij} = A_{ji}$

✅ Correct! Entry (1,3) of R^T equals entry (3,1) of R, which is 4.

❌ Not quite. Remember: to find (R^T)ᵢⱼ, look at Rⱼᵢ — swap the row and column indices.

Show solution

Solution:

We need entry $(1, 3)$ of $R^T$ , meaning row 1, column 3.

By the transpose definition: $(R^T)_{1,3} = R_{3,1}$

Looking at $R$ : row 3, column 1 is 4.

$( R^T )_{1,3} = R_{3,1} = 4$

Question 2

What is $(AB)^T$ equal to?

✅ Correct! The reversal rule flips the order: (AB)^T = B^T A^T.

❌ Not quite. Remember the socks-and-shoes rule — transposing a product reverses the order.

Show solution

Solution:

The reversal rule states:

$(AB)^T = B^T A^T$

The order reverses! Think of the socks-and-shoes analogy:

Put on: socks first, then shoes $\to AB$
Take off: shoes first, then socks $\to B^T A^T$

Common mistake: Writing $A^T B^T$ keeps the same order — this is wrong!

Question 3

True or False: If $R$ is a $3 \times 2$ matrix, then $R^T R$ is a $3 \times 3$ matrix.

✅ Correct! R^T R is 2×2, not 3×3. The product (2×3)(3×2) gives a 2×2 result.

❌ Not quite. If R is 3×2, then R^T is 2×3. Multiply the dimensions: (2×3)(3×2) = ?

Show solution

Solution: False

If $R$ is $3 \times 2$ , then $R^T$ is $2 \times 3$ .

$R^T R: \quad \underbrace{(2 \times 3)}_{R^T} \cdot \underbrace{(3 \times 2)}_{R} = \underbrace{2 \times 2}_{\text{result}}$

The inner dimensions (3 and 3) match for multiplication, and the outer dimensions (2 and 2) give the result size.

It's $R R^T$ that would be $3 \times 3$ :

$R R^T: \quad \underbrace{(3 \times 2)}_{R} \cdot \underbrace{(2 \times 3)}_{R^T} = \underbrace{3 \times 3}_{\text{result}}$

Question 4

Using $R = \begin{bmatrix} 1 & 3 \\ 2 & 3 \\ 4 & 1 \end{bmatrix}$ , what is the $(1, 2)$ entry of $R^T R$ ?

Hint: $(R^T R)_{1,2}$ is the dot product of column 1 of $R$ with column 2 of $R$ .

✅ Correct! 1·3 + 2·3 + 4·1 = 3 + 6 + 4 = 13. And by symmetry, the (2,1) entry is also 13!

❌ Not quite. Dot column 1 = (1, 2, 4) with column 2 = (3, 3, 1): multiply corresponding entries and add them up.

Show solution

Solution:

$(R^T R)_{1,2}$ = (row 1 of $R^T$ ) $\cdot$ (column 2 of $R$ )

Row 1 of $R^T$ = column 1 of $R$ = $(1,\ 2,\ 4)$

Column 2 of $R$ = $(3,\ 3,\ 1)$

$(R^T R)_{1,2} = 1 \cdot 3 + 2 \cdot 3 + 4 \cdot 1 = 3 + 6 + 4 = 13$

This also equals $(R^T R)_{2,1} = 13$ — that's symmetry in action!

Solved: 0 / 4

Matrix Transposes and Symmetric Matrices

IKey Formulas

IIVisualization 1 — Rows Become Columns

IIIVisualization 2 — RTRR^T RRTR Is Always Symmetric

IVVisualization 3 — Same Symmetry, Different Sizes

VQuiz Questions

IIIVisualization 2 — $R^T R$ Is Always Symmetric