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Problem type

Determinant of a matrix

What it solves
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.
Subject
Linear Algebra
Topic
Determinants and inverses
Also called
det, cofactor expansion, Laplace expansion
Your problem

Find the determinant of this matrix.

det⁡[5−1331−2415]\det \left[ \begin{array}{rrr} 5 & -1 & 3 \\ 3 & 1 & -2 \\ 4 & 1 & 5 \end{array} \right]
The square matrix
Size3
Whole numbers or fractions. 2 by 2, 3 by 3 or 4 by 4.
Invent a new one:
Femtobytes
Determinant of the 3 by 3 matrix 5 -1 3 / 3 1 -2 / 4 1 5, step by step
femtobytes.com/solve/determinant/5c-1c3r3c1c-2r4c1c5
The problem

Find the determinant of this matrix.

det⁡[5−1331−2415]\det \left[ \begin{array}{rrr} 5 & -1 & 3 \\ 3 & 1 & -2 \\ 4 & 1 & 5 \end{array} \right]
The result
det⁡[5−1331−2415]=55\det \left[ \begin{array}{rrr} 5 & -1 & 3 \\ 3 & 1 & -2 \\ 4 & 1 & 5 \end{array} \right] = 55

The determinant is not 00, so the matrix has an inverse.

checked 32 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
55−1-1333311−2-2441155
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.

det⁡[5−1331−2415]\det \left[ \begin{array}{rrr} 5 & -1 & 3 \\ 3 & 1 & -2 \\ 4 & 1 & 5 \end{array} \right]
The quantity we are asked for.
What the determinant measures
55−1-1333311−2-2441155
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 3×33 \times 3 determinant is built from three 2×22 \times 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 11 and column 11. What is left is M1=[1−215]M_{1} = \left[ \begin{array}{rr} 1 & -2 \\ 1 & 5 \end{array} \right], and its determinant is:

det⁡M1\displaystyle \det M_{1}==1⋅5−(−2⋅1)\displaystyle 1 \cdot 5 - \left( -2 \cdot 1 \right)
For a 2×22 \times 2 matrix: the product down the main diagonal minus the product up the other diagonal. checked
==5−(−2)\displaystyle 5 - \left( -2 \right)
Do the two multiplications. checked
==7\displaystyle 7
Subtract. Subtracting a negative number adds. checked

Strike out row 11 and column 22. What is left is M2=[3−245]M_{2} = \left[ \begin{array}{rr} 3 & -2 \\ 4 & 5 \end{array} \right], and its determinant is:

det⁡M2\displaystyle \det M_{2}==3⋅5−(−2⋅4)\displaystyle 3 \cdot 5 - \left( -2 \cdot 4 \right)
For a 2×22 \times 2 matrix: the product down the main diagonal minus the product up the other diagonal. checked
==15−(−8)\displaystyle 15 - \left( -8 \right)
Do the two multiplications. checked
==23\displaystyle 23
Subtract. Subtracting a negative number adds. checked

Strike out row 11 and column 33. What is left is M3=[3141]M_{3} = \left[ \begin{array}{rr} 3 & 1 \\ 4 & 1 \end{array} \right], and its determinant is:

det⁡M3\displaystyle \det M_{3}==3⋅1−1⋅4\displaystyle 3 \cdot 1 - 1 \cdot 4
For a 2×22 \times 2 matrix: the product down the main diagonal minus the product up the other diagonal. checked
==3−4\displaystyle 3 - 4
Do the two multiplications. checked
==−1\displaystyle -1
Subtract. Subtracting a negative number adds. checked
Expanding along the first row
55−1-1333311−2-2441155
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

5

Pivot 1: column 1, row 1

6

Pivot 2: column 2, row 2

7

Pivot 3: column 3, row 3

8

Multiply down the diagonal