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Wize University Linear Algebra Textbook > Determinants

(Optional) Shortcut for 3x3 Determinants

Determinants of Large MatricesPrevious Section
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Shortcut for 3×3\colorOne{3 \times 3} Determinants

There is a quick and easy way of finding the determinant of a 3×33 \times 3 matrix called the Rule of Sarrus.
Steps
  1. Copy the first 2 columns to the right of the original matrix
  2. Draw diagonal arrows spanning three entries each
  3. Multiply along the arrows
  4. Products with arrows from top to bottom are added (++)
  5. Products with arrows from bottom to top are subtracted (−-)
Example
Given the matrix 𝐴=[123456789]𝐴=\begin{bmatrix} 1&2&3\\ 4&5&6\\ 7&8&9\\ \end{bmatrix}, find det(A)\det A.

det(A)=(1)(5)(9)+(2)(6)(7)+(3)(4)(8)−(7)(5)(3)−(8)(6)(1)−(9)(4)(2)=45+84+96−105−48−72=0\begin{aligned} \det A &=(1)(5)(9)+(2)(6)(7)+(3)(4)(8)−(7)(5)(3)−(8)(6)(1)−(9)(4)(2)\\[0.5em] &=45+84+96-105-48-72\\[0.5em] &= \boxed{0} \end{aligned}
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Example: Shortcut for 3×3\colorOne{3 \times 3} Determinants

A=[20−21−10504]A= \left[ \begin{array}{rrr} 2 & 0 & -2\\ 1 & -1 & 0\\ 5 & 0 & 4\\ \end{array} \right]. Use the Rule of Sarrus to find det(A)\det A.

det(A)=(2)(−1)(4)+(0)(0)(5)+(−2)(1)(0)−(5)(−1)(−2)−(0)(0)(2)−(4)(1)(0)=−8+0+0−10−0−0=−18\begin{aligned} \det A &= (2)(-1)(4) + (0)(0)(5) + (-2)(1)(0) - (5)(-1)(-2) - (0)(0)(2) - (4)(1)(0)\\ &= -8 + 0 + 0 -10 - 0 - 0\\ &= \boxed{-18} \end{aligned}

Let A=[−61aa−2001−2]A=\left[\begin{array}{rrr} -6&1& a\\ a&-2&0\\ 0&1& -2 \end{array}\right]. Find the value(s) of aa such that AA is not invertible.
Extra Practice
133 - FML 3 - 18.1W e.g. 71
  ⁣B‾=  [1234−112x2]\bcb{\boldsymbol{ \ul{B} =  \begin{bmatrix} 1 & 2 & 3 \\ 4 & -1 & 1 \\ 2 & x & 2 \end{bmatrix} }} such that det(  ⁣B‾)=−8 \bcb{\boldsymbol{ \det {\ul{B}} = -8 }} , then x\bcb{\boldsymbol{ x }} is:
(a)  111(b)  −1(c)  0(d)  None of the above\text{(a)}\;\boldsymbol{ \frac{1}{11} } \qquad\qquad\qquad \text{(b)}\; \boldsymbol{ -1}\qquad\qquad\qquad \text{(c)}\; \boldsymbol{ 0}\qquad\qquad\qquad \text{(d)}\; \text{\bf{None of the above}}
Shortcut for 3x3 Determinants
Let A=[−61aa−2001−2]A=\left[\begin{array}{rrr} -6&1& a\\ a&-2&0\\ 0&1& -2 \end{array}\right]. Find the value(s) of aa such that AA is not invertible.
Determinants of Large Matrices
B=[40151002−2]B= \begin{bmatrix} 4&0&1\\ 5&1&0\\ 0&2&-2 \end{bmatrix}
Find det(B)\det B.
Practice: Determinant of $3\times3$ Matrix
Let A=[61aa−20012]A=\left[\begin{array}{rrr} 6&1&a\\ a&-2&0\\ 0&1&2 \end{array}\right] using cofactor expansion along any row or column
Find the value(s) of aa that will make AA invertible
Practice: Determinant
Practice Question: Determinant
Find the determinant of [40151002−2]\begin{bmatrix} 4&0&1\\ 5&1&0\\ 0&2&-2 \end{bmatrix}.
Practice: Determinant
Practice: Determinant
Find the determinant of [40151002−2]\begin{bmatrix} 4&0&1\\ 5&1&0\\ 0&2&-2 \end{bmatrix}.