LINEAR-ALGEBRA · Unit 17 · Video 4 · Interactive Practice

How Determinants Turn Matrix Multiplication into Simple Arithmetic

IKey Formulas

Formula	Name	Description
$\det(AB) = \det(A) \cdot \det(B)$	Multiplicative Property (P9)	Products become scalar multiplication
$\det(A^{-1}) = \frac{1}{\det(A)}$	Inverse Determinant	Requires $\det(A) \neq 0$
$\det(cA) = c^n \cdot \det(A)$	Scalar Multiplication	$n$ = matrix size, $c$ = scalar
$\det(A^T) = \det(A)$	Transpose Property (P10)	Row properties $=$ column properties

IIVisualization 1 — Scaling by $c$ Multiplies the Determinant by $c^n$

Multiply every entry of an $n \times n$ matrix by $c$ and the determinant scales by $c^n$ , not by $c$ .

Scalar c 2 Dimension n

IIIVisualization 2 — Composing Transformations Multiplies Their Determinants

Apply $B$ , then $A$ : the area-scale factors multiply, so $\det(AB) = \det(A) \cdot \det(B)$ .

a₁₁ 1 a₁₂ 1 a₂₁ 0 a₂₂ 2 b₁₁ 2 b₁₂ 0 b₂₁ 1 b₂₂ 1

$\det$ has no addition rule: $\det(A + B) \neq \det(A) + \det(B)$ in general — e.g. $\det(2I) = 4$ but $\det(I) + \det(I) = 2$ .

IVVisualization 3 — Powers and Inverses Reuse One Number

One number fixes every power and inverse of $A$ : $\det(A^k) = [\det(A)]^k$ and $\det(A^{-1}) = 1/\det(A)$ .

det(A) 2

VQuiz Questions

Question 1

Let $A$ be a $4 \times 4$ matrix with $\det(A) = 3$ .

What is $\det(2A)$ ?

✅ Correct! Each of the 4 rows contributes a factor of 2, giving $2^4 \cdot 3 = 48$ .

❌ Not quite. Remember: $\det(cA) = c^n \cdot \det(A)$ where $n$ is the matrix dimension. Here $c = 2$ and $n = 4$ .

Show solution

Solution:

For a $4 \times 4$ matrix with $\det(A) = 3$ :

$\det(2A) = 2^4 \cdot \det(A) = 16 \cdot 3 = 48$

Common mistakes:

6: computed $2 \cdot 3$ — forgot that the scalar is raised to the power $n$
16: computed $2^4 = 16$ — forgot to multiply by $\det(A)$
24: used $2^3 \cdot 3 = 24$ — used the wrong dimension ( $n = 3$ instead of $n = 4$ )

Question 2

Let $A$ and $B$ both equal the $2 \times 2$ identity matrix $I_2$ . Which statement is correct?

✅ Correct! This classic counterexample proves that determinants do NOT respect addition.

❌ Not quite. Compute $\det(2I_2)$ using the scaling property: $\det(cA) = c^n \cdot \det(A)$ .

Show solution

Solution:

With $A = B = I_2$ :

$A + B = 2I_2 = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}$

$\det(A + B) = \det(2I_2) = 2^2 \cdot \det(I_2) = 4 \cdot 1 = 4$

$\det(A) + \det(B) = 1 + 1 = 2$

Since $4 \neq 2$ , the determinant does not distribute over addition. This is the classic counterexample from the video!

Question 3

If $\det(A) = 5$ , what is $\det(A^3) \cdot \det(A^{-1})$ ?

Hint: Use both the power rule and the inverse rule.

✅ Correct! $5^3 \cdot \frac{1}{5} = 25$ , or equivalently $\det(A^2) = 5^2 = 25$ .

❌ Not quite. Break it into parts: $\det(A^3) = [\det(A)]^3$ and $\det(A^{-1}) = 1/\det(A)$ . Then multiply.

Show solution

Solution:

Using the power rule and the inverse rule:

$\det(A^3) = [\det(A)]^3 = 5^3 = 125$

$\det(A^{-1}) = \frac{1}{\det(A)} = \frac{1}{5}$

$\det(A^3) \cdot \det(A^{-1}) = 125 \cdot \frac{1}{5} = 25$

Elegant alternative: By the multiplicative property:

$\det(A^3) \cdot \det(A^{-1}) = \det(A^3 \cdot A^{-1}) = \det(A^2) = [\det(A)]^2 = 5^2 = 25$

Question 4

True or False: If swapping two rows of a matrix flips the sign of the determinant, then swapping two columns also flips the sign of the determinant.

✅ Correct! The transpose property guarantees that every row property has an automatic column counterpart.

❌ Not quite. Think about what $\det(A^T) = \det(A)$ implies: a column operation on $A$ equals the corresponding row operation on $A^T$ .

Show solution

Solution: True

By the transpose property, $\det(A^T) = \det(A)$ .

A column operation on $A$ is the same as the corresponding row operation on $A^T$ . Since $\det(A^T) = \det(A)$ , any property that holds for rows automatically holds for columns.

Specifically: Swapping columns $i$ and $j$ of $A$ is the same as swapping rows $i$ and $j$ of $A^T$ . The row swap flips the sign of $\det(A^T)$ . But $\det(A^T) = \det(A)$ , so the column swap flips the sign of $\det(A)$ too.

Every row rule has a column twin — for free!

Solved: 0 / 4

How Determinants Turn Matrix Multiplication into Simple Arithmetic

IKey Formulas

IIVisualization 1 — Scaling by ccc Multiplies the Determinant by cnc^ncn

IIIVisualization 2 — Composing Transformations Multiplies Their Determinants

IVVisualization 3 — Powers and Inverses Reuse One Number

VQuiz Questions

IIVisualization 1 — Scaling by $c$ Multiplies the Determinant by $c^n$