01 Key Concepts
Determinant of a 2x2 Matrix
For [[a,b],[c,d]], the determinant equals a*d - b*c.
Notation
The determinant of matrix A is written det(A) or with vertical bars around the matrix entries.
What the Determinant Tells You
If det(A) = 0, the matrix is 'singular' and has no inverse. If det(A) is not 0, the matrix does have an inverse.
Determinant of a 3x3 Matrix
Calculated using cofactor expansion, breaking the 3x3 determinant down into a combination of three 2x2 determinants.
Determinants and Area
For a 2x2 matrix formed from two vectors, the absolute value of the determinant equals the area of the parallelogram those vectors form.
02 Key Formulas
- det([[a,b],[c,d]]) = a*d - b*c
03 Solved Examples
Example 1 Find the determinant of [[3,2],[1,4]].
- Apply the formula: a*d - b*c = 3*4 - 2*1.
- = 12 - 2.
Answer: det = 10
Example 2 Find the determinant of [[2,4],[1,2]] and determine if it's invertible.
- Apply the formula: 2*2 - 4*1 = 4 - 4.
Answer: det = 0, so this matrix is NOT invertible (it's singular)
Example 3 Find the determinant of [[5,0],[0,5]].
- Apply the formula: 5*5 - 0*0.
Answer: det = 25
04 Practice Questions
1Find the determinant of [[1,2],[3,4]].
-2
2Find the determinant of [[6,3],[2,1]].
0
3Is a matrix with determinant 0 invertible?
No
4Find the determinant of [[7,0],[0,3]].
21
5Find the determinant of [[2,5],[1,3]].
1
📄 Determinants — Downloadable Worksheet
10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.