For [[a, b], [c, d]] the determinant is ad − bc. It is a single number that summarises the whole matrix.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
What it measures
The determinant is the area scaling factor of the transformation. A determinant of 3 triples every area.
A determinant of zero
Zero means the transformation flattens the plane onto a line, destroying information. Such a matrix is called singular and has no inverse.
The 2×2 inverse
Swap a and d, negate b and c, and divide everything by the determinant. The division is why a zero determinant blocks it.
The identity
A matrix times its inverse gives the identity matrix, which has 1s down the diagonal and 0s elsewhere and changes nothing.
Larger matrices
A 3×3 determinant expands along a row, taking each entry times the determinant of the 2×2 left when its row and column are deleted, with alternating signs.
One number that decides invertibility
The determinant of a square matrix is a single number. If it is zero the matrix has no inverse; otherwise it does. That one test settles whether a system has a unique solution.
It measures area scaling
The determinant tells you the factor by which the transformation scales area, or volume in three dimensions. A determinant of zero means the transformation collapses everything onto a line, which is why it cannot be undone.
The inverse undoes the transformation
A matrix times its inverse gives the identity, which leaves every vector unchanged. Finding the inverse for a 2×2 matrix is a short formula; for larger ones it is systematic elimination.
A negative determinant flips orientation
The sign records whether the transformation reverses orientation, as a reflection does. So the determinant carries both a magnitude and a handedness, which is more information than it first appears to hold.
Step 2: Try It Yourself
Tap and try it out.
- Determinant of M4
A non-zero determinant means the matrix is invertible. Its size is the factor by which areas are scaled — here 4.
Step 3: Watch an Example
One step at a time.
Watch Sana Invert [[4, 7], [2, 6]]
Sana needs the inverse of this matrix.
- Step 1
The determinant is 4×6 − 7×2 = 24 − 14 = 10.
Step 4: Your Turn
Practice makes it stick.
The Determinant
Problem 1 of 2
[[3, 1], [2, 4]]. What is the determinant?
The Singular One
Problem 2 of 2
[[2, 4], [1, 2]]. What is the determinant?
Determine and Invert
1 of 8
[[5, 2], [3, 4]]. Determinant?
2 of 8
[[1, 0], [0, 1]]. Determinant?
3 of 8
[[6, 3], [4, 2]]. Determinant?
4 of 8
[[2, 0], [0, 3]]. Determinant?
5 of 8
A determinant of 0 means an inverse exists? Enter 1 for yes, 0 for no.
6 of 8
A transformation has determinant 5. A shape of area 3 becomes what area?
7 of 8
Put the steps for finding a 2×2 inverse in order.
- 1Check that it is not zero.
- 2Swap a and d, and negate b and c.
- 3Divide every entry by the determinant.
- 4Compute the determinant, ad − bc.
8 of 8
[[7, 3], [2, 1]]. Determinant?
Step 5: Quick Check
Show what you know.
Question 1 of 2
[[4, 1], [3, 2]]. What is the determinant?
Question 2 of 2
What does a determinant of zero mean?
What You Learned
- The 2×2 determinant is ad − bc, and it measures area scaling.
- A determinant of zero means the matrix is singular and cannot be inverted.
- The 2×2 inverse swaps the diagonal, negates the off-diagonal, and divides by the determinant.