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Linear Algebra · Systems & Elimination

Determinants — Practice Problems and Worked Solutions

Track scaling, orientation, and invertibility.

The formulas you need here

Determinant of a 2×2

det (a c b d) = a d - b c

What it says: The signed factor by which the matrix scales area.
When to reach for it: Testing invertibility, and as the denominator of the 2×2 inverse.
Watch out: Determinant zero means singular — no inverse, and the transformation flattens space.

det (2412) = 0

The rows are proportional, so the matrix is not invertible.

18 worked problems

Every step is generated and checked by a computer algebra system.

1.
Compute the determinant.
$det ([1324])$
Answer: $- 2$
Step-by-step solution
1. $E x p an d a l o n g a co n v e ni e n t r o w or u see l imina t i o n .$
2. $det (A) = - 2$
2.
Compute the determinant.
$det ([25 - 1 3])$
Answer: $11$
Step-by-step solution
1. $E x p an d a l o n g a co n v e ni e n t r o w or u see l imina t i o n .$
2. $det (A) = 11$
3.
Compute the determinant.
$det ([4 - 2 06])$
Answer: $24$
Step-by-step solution
1. $E x p an d a l o n g a co n v e ni e n t r o w or u see l imina t i o n .$
2. $det (A) = 24$
4.
Compute the determinant.
$det 102230011$
Answer: $7$
Step-by-step solution
1. $E x p an d a l o n g a co n v e ni e n t r o w or u see l imina t i o n .$
2. $det (A) = 7$
5.
Compute the determinant.
$det 201 - 1 40 325$
Answer: $26$
Step-by-step solution
1. $E x p an d a l o n g a co n v e ni e n t r o w or u see l imina t i o n .$
2. $det (A) = 26$
6.
Compute the determinant.
$det 32 - 1 142 - 1 05$
Answer: $42$
Step-by-step solution
1. $E x p an d a l o n g a co n v e ni e n t r o w or u see l imina t i o n .$
2. $det (A) = 42$
7.
Compute the determinant.
$det ([2134])$
Answer: $5$
Step-by-step solution
1. $E x p an d a l o n g a co n v e ni e n t r o w or u see l imina t i o n .$
2. $det (A) = 5$
8.
Compute the determinant.
$det ([53 - 2 1])$
Answer: $11$
Step-by-step solution
1. $E x p an d a l o n g a co n v e ni e n t r o w or u see l imina t i o n .$
2. $det (A) = 11$
9.
Compute the determinant.
$det ([6 - 1 02])$
Answer: $12$
Step-by-step solution
1. $E x p an d a l o n g a co n v e ni e n t r o w or u see l imina t i o n .$
2. $det (A) = 12$
10.
Compute the determinant.
$det ([1253])$
Answer: $- 7$
Step-by-step solution
1. $E x p an d a l o n g a co n v e ni e n t r o w or u see l imina t i o n .$
2. $det (A) = - 7$
11.
Compute the determinant.
$det ([4327])$
Answer: $22$
Step-by-step solution
1. $E x p an d a l o n g a co n v e ni e n t r o w or u see l imina t i o n .$
2. $det (A) = 22$
12.
Compute the determinant.
$det 210032104$
Answer: $26$
Step-by-step solution
1. $E x p an d a l o n g a co n v e ni e n t r o w or u see l imina t i o n .$
2. $det (A) = 26$
13.
Compute the determinant.
$det 132 - 1 04 210$
Answer: $18$
Step-by-step solution
1. $E x p an d a l o n g a co n v e ni e n t r o w or u see l imina t i o n .$
2. $det (A) = 18$
14.
Compute the determinant.
$det 310215042$
Answer: $- 58$
Step-by-step solution
1. $E x p an d a l o n g a co n v e ni e n t r o w or u see l imina t i o n .$
2. $det (A) = - 58$
15.
Compute the determinant.
$det ([7215])$
Answer: $33$
Step-by-step solution
1. $E x p an d a l o n g a co n v e ni e n t r o w or u see l imina t i o n .$
2. $det (A) = 33$
16.
Compute the determinant.
$det 124035006$
Answer: $18$
Step-by-step solution
1. $E x p an d a l o n g a co n v e ni e n t r o w or u see l imina t i o n .$
2. $det (A) = 18$
17.
Compute the determinant.
$det ([2164])$
Answer: $2$
Step-by-step solution
1. $E x p an d a l o n g a co n v e ni e n t r o w or u see l imina t i o n .$
2. $det (A) = 2$
18.
Compute the determinant.
$det 301320154$
Answer: $37$
Step-by-step solution
1. $E x p an d a l o n g a co n v e ni e n t r o w or u see l imina t i o n .$
2. $det (A) = 37$

The formulas you need here

Determinant of a 2×2

18 worked problems

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