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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⁡[−1−520]\det \left[ \begin{array}{rr} -1 & -5 \\ 2 & 0 \end{array} \right]
The square matrix
Size2
Whole numbers or fractions. 2 by 2, 3 by 3 or 4 by 4.
Invent a new one:
Femtobytes
Determinant of the 2 by 2 matrix -1 -5 / 2 0, step by step
femtobytes.com/solve/determinant/-1c-5r2c0
The problem

Find the determinant of this matrix.

det⁡[−1−520]\det \left[ \begin{array}{rr} -1 & -5 \\ 2 & 0 \end{array} \right]
The result
det⁡[−1−520]=10\det \left[ \begin{array}{rr} -1 & -5 \\ 2 & 0 \end{array} \right] = 10

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

checked 3 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
−1-1−5-52200
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⁡[−1−520]\det \left[ \begin{array}{rr} -1 & -5 \\ 2 & 0 \end{array} \right]
The quantity we are asked for.
What the determinant measures
−1-1−5-52200
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

Apply the two by two rule

Why this step

For a 2×22 \times 2 matrix the area factor can be worked out once and for all, and it comes to one product minus another. The columns of the matrix are the two sides of a parallelogram; the rule is the area of that parallelogram, with a sign for orientation.

det⁡A\displaystyle \det A==−1⋅0−(−5⋅2)\displaystyle -1 \cdot 0 - \left( -5 \cdot 2 \right)
For a 2×22 \times 2 matrix: the product down the main diagonal minus the product up the other diagonal. checked
==0−(−10)\displaystyle 0 - \left( -10 \right)
Do the two multiplications. checked
==10\displaystyle 10
Subtract. Subtracting a negative number adds. checked
-7-6-5-4-3-2-101-3-2-1012345xyarea 10first column (-1, 2)second column (-5, 0)the parallelogram they span, area 10
The two columns of the matrix drawn as arrows from the origin, and the parallelogram they span. Its area is 10, the size of the determinant. The determinant is positive because turning from the first arrow to the second goes anticlockwise.
The matrix whose determinant is wanted
−1-1−5-52200
For a two by two matrix the determinant is one subtraction: the two entries on the main diagonal multiplied, less the two on the other diagonal.