Properties of determinants

Computing a determinant by direct expansion is feasible for $2 \times 2$ and $3 \times 3$ matrices, but painful for larger ones. The seven properties listed below let you simplify the determinant first , often reducing it to an obvious value with little arithmetic. These properties are also the basis of nearly every JEE-style determinant question.

The seven properties

P1. $\det A^T = \det A$. The determinant is unchanged by transposition.

Consequence. Every property stated for rows holds also for columns.

P2. If two rows (or columns) of $A$ are interchanged, $\det A$ changes sign.

Consequence. If two rows are equal, $\det A = 0$ (swapping them gives both the original and its negative).

P3. Multiplying a row (or column) by a scalar $\lambda$ multiplies the determinant by $\lambda$.

Consequence. $\det(\lambda A) = \lambda^n \det A$ for an $n \times n$ matrix.

P4. $\det A = 0$ if any row (or column) is entirely zero.

P5. Adding a multiple of one row to another does not change the determinant. Same for columns.

Reason. The contribution of the added multiple cancels because of P2 , adding a row to itself would create two equal rows.

P6. Sum-of-rows: if row $i$ of $A$ is a sum $u + v$, then $\det A = \det A_u + \det A_v$, where $A_u, A_v$ have row $i$ equal to $u$ and $v$ respectively, with all other rows the same.

P7. $\det(AB) = \det A \cdot \det B$. Multiplicative property for matrix products.

Two more useful consequences

• If any row is a scalar multiple of another row, $\det A = 0$.
• $\det(A^{-1}) = 1 / \det A$ for invertible $A$.

Strategic use of properties

Pattern: extracting a common factor. If a row has a common factor, pull it out using P3.

Pattern: simplifying via row addition. Replace a row by itself plus a multiple of another (P5). The determinant is unchanged but the matrix may simplify.

Pattern: detecting linear dependence. If $R_3 = a R_1 + b R_2$, then $\det = 0$. This is often hidden , look carefully.

A long worked example

Compute $\det\begin{pmatrix} a & a + b & a + 2b \\ b & b + c & b + 2c \\ c & c + d & c + 2d \end{pmatrix}$.

$C_2 \to C_2 - C_1$ (P5): $\begin{pmatrix} a & b & a + 2b \\ b & c & b + 2c \\ c & d & c + 2d \end{pmatrix}$.

$C_3 \to C_3 - C_1$ (P5): $\begin{pmatrix} a & b & 2b \\ b & c & 2c \\ c & d & 2d \end{pmatrix}$.

$C_3 = 2 C_2$ (every entry double). So columns are linearly dependent: $\det = 0$.

Vandermonde determinant

$\det\begin{pmatrix} 1 & 1 & 1 \\ a & b & c \\ a^2 & b^2 & c^2 \end{pmatrix} = (b - a)(c - a)(c - b)$.

Sketch of proof. $C_2 \to C_2 - C_1$, $C_3 \to C_3 - C_1$, then expand along the first row. Each column factors as $(b - a)$ or $(c - a)$. The remaining $2 \times 2$ determinant gives $(c - b)$.

This identity generalises: an $n \times n$ Vandermonde matrix has determinant $\prod_{i < j}(x_j - x_i)$.

Worked examples

Example 1. Without expanding, show $\det\begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 5 & 6 & 7 \end{pmatrix} = 0$.

Row $2 = 2 R_1$. So determinant is zero.

Example 2. Show $\det\begin{pmatrix} 1 & 1 & 1 \\ x & y & z \\ x^2 & y^2 & z^2 \end{pmatrix} = (y - x)(z - x)(z - y)$.

Vandermonde formula.

Example 3. Find the determinant of $\begin{pmatrix} 2 & 3 & 1 \\ 4 & 6 & 2 \\ 6 & 9 & 3 \end{pmatrix}$.

Row $2 = 2 R_1$, row $3 = 3 R_1$. So determinant is zero.

Example 4. Show $\det\begin{pmatrix} 2 & 4 & 6 \\ 1 & 3 & 5 \\ 0 & 2 & 4 \end{pmatrix} = 0$.

$R_1 = 2 R_2 + 0 \cdot R_3$ no...let me check: $2 R_2 = (2, 6, 10)$, not $R_1$. Try $R_1 - 2 R_2 = (0, -2, -4)$. Then $R_1 - 2R_2 = -R_3$. So $R_1, R_2, R_3$ are linearly dependent, and the determinant is zero.

Example 5. Prove $\det\begin{pmatrix} a - b - c & 2a & 2a \\ 2b & b - c - a & 2b \\ 2c & 2c & c - a - b \end{pmatrix} = (a + b + c)^3$.

$R_1 \to R_1 + R_2 + R_3$: top row becomes $(a + b + c, a + b + c, a + b + c)$. Factor out $a + b + c$:

$(a + b + c) \det\begin{pmatrix} 1 & 1 & 1 \\ 2b & b - c - a & 2b \\ 2c & 2c & c - a - b \end{pmatrix}$.

$C_2 \to C_2 - C_1$, $C_3 \to C_3 - C_1$: $\begin{pmatrix} 1 & 0 & 0 \\ 2b & -(a + b + c) & 0 \\ 2c & 0 & -(a + b + c) \end{pmatrix}$.

Triangular, determinant $= 1 \cdot -(a + b + c) \cdot -(a + b + c) = (a + b + c)^2$. Multiplied by the factored $(a + b + c)$: $(a + b + c)^3$. $\checkmark$

Example 6. Show $\det\begin{pmatrix} a & a + 1 & a + 2 \\ b & b + 1 & b + 2 \\ c & c + 1 & c + 2 \end{pmatrix} = 0$.

$C_2 \to C_2 - C_1, C_3 \to C_3 - C_1$: $\begin{pmatrix} a & 1 & 2 \\ b & 1 & 2 \\ c & 1 & 2 \end{pmatrix}$. Columns $2$ and $3$ are proportional: $C_3 = 2 C_2$. So determinant $= 0$.

Try it yourself

1. Without expanding, show $\det\begin{pmatrix} 3 & 6 & 9 \\ 1 & 2 & 3 \\ 5 & 10 & 15 \end{pmatrix} = 0$.
2. Show that $\det A^T = \det A$ for $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ by direct computation.
3. If $\det A = 5$ for a $3 \times 3$ matrix $A$, find $\det(2A)$.
4. If $\det A = 3, \det B = 4$, find $\det(AB)$.
5. If $\det A = 3$, find $\det A^{-1}$.
6. Compute $\det\begin{pmatrix} 1 & a & a^2 \\ 1 & b & b^2 \\ 1 & c & c^2 \end{pmatrix}$ using Vandermonde.
7. Show that if a row of $A$ is a linear combination of other rows, then $\det A = 0$.
8. Compute $\det\begin{pmatrix} 1 + x & 1 & 1 \\ 1 & 1 + y & 1 \\ 1 & 1 & 1 + z \end{pmatrix}$.
9. Without expansion, show $\det\begin{pmatrix} a & b & c \\ b & c & a \\ c & a & b \end{pmatrix} = -(a^3 + b^3 + c^3 - 3abc)$.
10. Show $\det(\lambda A) = \lambda^n \det A$ for an $n \times n$ matrix.
11. Prove: $\det(A^k) = (\det A)^k$.
12. Show that the determinant of a permutation matrix is $\pm 1$.
13. Use properties to compute $\det\begin{pmatrix} 1 & 1 & 1 \\ a & b & c \\ bc & ca & ab \end{pmatrix}$.
14. Show $\det\begin{pmatrix} \sin\alpha & \cos\alpha & \sin(\alpha + \delta) \\ \sin\beta & \cos\beta & \sin(\beta + \delta) \\ \sin\gamma & \cos\gamma & \sin(\gamma + \delta) \end{pmatrix} = 0$.

Pitfalls / Tricks

• $\det(A + B) \neq \det A + \det B$ in general. Be careful.
• $\det(\lambda A) = \lambda^n \det A$, not $\lambda \det A$.
• A common factor in a row gives a multiplicative factor; a common factor in the matrix gives the $n$-th power.
• $\det A = 0$ ⟺ $A$ is singular ⟺ rows linearly dependent.
• Always inspect for obvious patterns before computing. A wasted expansion is the price of inattention.

Next we use determinants for a geometric application.