Properties of determinants

See also: Linear Algebra, Special Matrices, column space, null space, solution space

1. The Multiplicative Property (The “Powerhouse”)

This is the single most useful property for algebraic proofs involving determinants.

$\det (AB)=\det (A)\det (B)$

This means the determinant of a product is the product of the determinants.

• Why it matters: It lets you break complex matrix equations into simple scalar equations.

• Corollary (Powers): If you multiply $A$ by itself $k$ times:

  $\det (A^k)=(\det (A){)}^k$

  This is exactly the rule used in your $A^2=A$ example.

2. The Inverse Property

If $A$ is invertible (nonsingular), then:

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

• Derivation:

  $A{A}^{-1}=I$

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

  $\det (A)\det (A^{-1})=1(\text{Since }\det (I)=1)$

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

3. The Scalar Multiplication “Trap”

This is the most common mistake students make. If you multiply the entire matrix $A$ (size $n\times n$) by a scalar constant $k$:

$\det (kA)=k^n\det (A)$

• Why $k^n$ and not just $k$?

  Recall that the determinant measures volume. If you scale a 3D box ($n=3$) by 2 in every direction ($k=2$), the volume doesn’t double; it gets multiplied by $2\times 2\times 2=8$.

  • Example: If $\det (A)=3$ and $A$ is $3\times 3$, then $\det (2A)=2^3\cdot 3=8\cdot 3=24$.

4. The Transpose Property

$\det (A^T)=\det (A)$

This means “rows and columns are equal” in the eyes of the determinant. Any rule that applies to rows (like row operations) also applies to columns.

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Operation	Property	Note
Product	$\det (AB)=\det (A)\det (B)$	Determinant of product = product of determinants.
Power	$\det (A^k)=(\det (A){)}^k$	Follows from product rule.
Inverse	$\det (A^{-1})=\frac{1}{\det (A)}$	Only if $\det (A)\ne 0$.
Transpose	$\det (A^T)=\det (A)$	Rows vs Columns doesn’t matter for det.
Scalar Scale	$\det (kA)=k^n\det (A)$	Crucial: $n$ is the dimension of the matrix.
Identity	$\det (I)=1$	The unit volume.
Orthogonal	$\det (Q)=\pm 1$	Rotations ($+1$) or Reflections ($-1$).

Would you like to try a “trick question” involving the Scalar Scale property ($\det (kA)$) to make sure you’ve mastered it?