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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⁡[21113−1−110−20−1−1−1−13]\det \left[ \begin{array}{rrrr} 2 & 1 & 1 & 1 \\ 3 & -1 & -1 & 1 \\ 0 & -2 & 0 & -1 \\ -1 & -1 & -1 & 3 \end{array} \right]
The square matrix
Size4
Whole numbers or fractions. 2 by 2, 3 by 3 or 4 by 4.
Invent a new one:
Femtobytes
Determinant of the 4 by 4 matrix 2 1 1 1 / 3 -1 -1 1 / 0 -2 0 -1 / -1 -1 -1 3, step by step
femtobytes.com/solve/determinant/2c1c1c1r3c-1c-1c1r0c-2c0c-1r-1c-1c-1c3
The problem

Find the determinant of this matrix.

det⁡[21113−1−110−20−1−1−1−13]\det \left[ \begin{array}{rrrr} 2 & 1 & 1 & 1 \\ 3 & -1 & -1 & 1 \\ 0 & -2 & 0 & -1 \\ -1 & -1 & -1 & 3 \end{array} \right]
The result
det⁡[21113−1−110−20−1−1−1−13]=−36\det \left[ \begin{array}{rrrr} 2 & 1 & 1 & 1 \\ 3 & -1 & -1 & 1 \\ 0 & -2 & 0 & -1 \\ -1 & -1 & -1 & 3 \end{array} \right] = -36

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

checked 31 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
2211111133−1-1−1-11100−2-200−1-1−1-1−1-1−1-133
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⁡[21113−1−110−20−1−1−1−13]\det \left[ \begin{array}{rrrr} 2 & 1 & 1 & 1 \\ 3 & -1 & -1 & 1 \\ 0 & -2 & 0 & -1 \\ -1 & -1 & -1 & 3 \end{array} \right]
The quantity we are asked for.
What the determinant measures
2211111133−1-1−1-11100−2-200−1-1−1-1−1-1−1-133
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

Reduce to a triangle

Why this step

Expanding a 4×44 \times 4 determinant along a row means four 3×33 \times 3 determinants, each of which means three 2×22 \times 2 ones: about forty multiplications. Row reduction gets there in a handful of moves, because for a triangular matrix (zeros below the diagonal) the determinant is just the product of the diagonal.

The matrix this second method starts from
2211111133−1-1−1-11100−2-200−1-1−1-1−1-1−1-133
The same matrix again. Row reduction will drive every number below the diagonal to zero; for a matrix like that the determinant is just the diagonal multiplied together, so the work is in getting there rather than in the answer.
3

Pivot 1: column 1, row 1

4

Pivot 2: column 2, row 2

5

Pivot 3: column 3, row 3

6

Pivot 4: column 4, row 4

7

Multiply down the diagonal