Find the determinant of a 2 by 2, 3 by 3 or 4 by 4 matrix: by cofactor expansion with every small determinant worked out, and by row reduction to a triangle, with the two answers checked against each other.
The determinant is not 0, so the matrix has an inverse.
checked32 lines of this solution were evaluated with exact arithmetic before it was shown to you. A solution with a false line is never served.
Picture 1 of 2Step 1: Say what the determinant measures
What the determinant measures
4−4231−3555
The rows of this matrix, read as arrows from the origin, span a box. The determinant is how much that box is scaled when the matrix acts, and a determinant of zero means the box has been flattened to nothing.
1
Say what the determinant measures
Why this step
A formula for the determinant can be memorised in a minute and forgotten in a week. What stays is knowing what the number MEANS, because the meaning explains every rule about it: why a swap flips its sign, why a zero determinant means no inverse, why it multiplies when matrices multiply.
det435−4152−35
The quantity we are asked for.
What the determinant measures
4−4231−3555
The rows of this matrix, read as arrows from the origin, span a box. The determinant is how much that box is scaled when the matrix acts, and a determinant of zero means the box has been flattened to nothing.
2
Expand along the first row
Why this step
A 3×3 determinant is built from three 2×2 ones, which we already know how to do. Each entry of the first row is paired with the small determinant left when its own row and column are struck out. The signs alternate +,−,+; they are not decoration: they are what makes the answer flip sign when two rows are swapped, as any measure of orientation must.
Strike out row 1 and column 1. What is left is M1=[15−35], and its determinant is:
detM1=1⋅5−(−3⋅5)
For a 2×2 matrix: the product down the main diagonal minus the product up the other diagonal. checked
=5−(−15)
Do the two multiplications. checked
=20
Subtract. Subtracting a negative number adds. checked
Strike out row 1 and column 2. What is left is M2=[35−35], and its determinant is:
detM2=3⋅5−(−3⋅5)
For a 2×2 matrix: the product down the main diagonal minus the product up the other diagonal. checked
=15−(−15)
Do the two multiplications. checked
=30
Subtract. Subtracting a negative number adds. checked
Strike out row 1 and column 3. What is left is M3=[3515], and its determinant is:
detM3=3⋅5−1⋅5
For a 2×2 matrix: the product down the main diagonal minus the product up the other diagonal. checked
=15−5
Do the two multiplications. checked
=10
Subtract. Subtracting a negative number adds. checked
Expanding along the first row
4−4231−3555
The first row lit. Each of its entries is taken in turn, and with it the smaller matrix that remains when its row and its column are covered up.
3
Combine the three pieces with their signs
4
Check by a completely different method: reduce to a triangle