Determinant Facts for Kids

A determinant is a special number that comes from a square matrix. Think of it as a secret code that tells you how the matrix acts. You can figure out this number using the numbers inside the matrix. We write the determinant of a matrix A as $\det(A)$ or $|A|$ .

What is a Singular Matrix?
How to Calculate a Determinant
- Formulas for Small Matrices
- Cofactor Expansion Method
Related pages
See also

What is a Singular Matrix?

A matrix can have an inverse matrix only if its determinant is not zero. If the determinant is not zero, the matrix is called invertible. This means you can "undo" what the matrix does.

If the determinant is zero, the matrix is called non-invertible or singular. A singular matrix "flattens" things. Imagine a 3D box becoming a flat 2D shape, or a 2D square becoming a straight line. When this happens, the "volume" or "area" becomes zero. You can't easily get the original shape back.

How to Calculate a Determinant

There are different ways to find a determinant. Here are some common methods.

Formulas for Small Matrices

The Rule of Sarrus helps you remember how to find the determinant of a 3x3 matrix. You multiply numbers along certain diagonals and then add or subtract them.

For very small matrices, the formulas are simple:

For a 1x1 matrix (just one number):

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \det\begin{bmatrix}a\end{bmatrix} = a

For a 2x2 matrix:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \det\begin{bmatrix}a&b\\c&d\end{bmatrix} = ad-bc You multiply the numbers on the main diagonal (Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): a \times d ) and subtract the product of the numbers on the other diagonal (Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): b \times c ).

For a 3x3 matrix, the formula is a bit longer:

${\det\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix} = {\color{blue}{aei}+{dhc}+{gbf}}{\color{OrangeRed}{}-{gec}-{ahf}-{dbi}}}$ You can use the Rule of Sarrus (look at the image) to help you remember this formula. It involves multiplying numbers along specific diagonal lines.

Cofactor Expansion Method

For bigger matrices, finding the determinant is more complex. One common way is called cofactor expansion.

Let's say you have an $n \times n$ matrix (meaning it has $n$ rows and $n$ columns). 1. First, pick any row or column in the matrix. 2. For each number $a_{ij}$ in that row or column, you need to find its cofactor $C_{ij}$ . 3. The determinant of the matrix is the sum of each number multiplied by its cofactor. So, $\det(A) = \sum a_{ij} C_{ij}$ .

To find a cofactor $C_{ij}$ : 1. Remove (or "erase") the row $i$ and column $j$ that the number $a_{ij}$ is in. 2. This leaves you with a smaller matrix, called $M$ . 3. The cofactor $C_{ij}$ is then $(-1)^{i+j} \det(M)$ . The Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): (-1)^{i+j} part means you multiply by 1 or -1 depending on if Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): i+j is even or odd.

Here is an example using cofactor expansion on the first column of a 3x3 matrix: Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \begin{align} \det \begin{bmatrix}{\color{red}1}&3&2\\{\color{red}2}&1&1\\{\color{red}0}&3&4\end{bmatrix} \end{align} We pick the first column (numbers are 1, 2, and 0).

For the number 1 (in row 1, column 1):

* Remove row 1 and column 1. The remaining matrix is Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \begin{bmatrix}1&1\\3&4\end{bmatrix} . * Its determinant is Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): (1 \times 4) - (1 \times 3) = 4 - 3 = 1 . * The cofactor is Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): (-1)^{1+1} \times 1 = 1 \times 1 = 1 .

For the number 2 (in row 2, column 1):

* Remove row 2 and column 1. The remaining matrix is Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \begin{bmatrix}3&2\\3&4\end{bmatrix} . * Its determinant is Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): (3 \times 4) - (2 \times 3) = 12 - 6 = 6 . * The cofactor is Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): (-1)^{2+1} \times 6 = -1 \times 6 = -6 .

For the number 0 (in row 3, column 1):

* Remove row 3 and column 1. The remaining matrix is Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \begin{bmatrix}3&2\\1&1\end{bmatrix} . * Its determinant is Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): (3 \times 1) - (2 \times 1) = 3 - 2 = 1 . * The cofactor is Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): (-1)^{3+1} \times 1 = 1 \times 1 = 1 .

Now, we add them up: Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \begin{align} &= ({\color{red}1} \cdot 1) + ({\color{red}2} \cdot (-6)) + ({\color{red}0} \cdot 1) \\ &= 1 - 12 + 0 \\ &= -11. \end{align}

A helpful trick: If you choose a row or column that has many zeros, the calculation becomes much easier. This is because if $a_{ij}$ is 0, then Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): a_{ij} \times C_{ij} will also be 0, so you don't need to calculate that cofactor.

Determinant facts for kids

Contents

What is a Singular Matrix?

How to Calculate a Determinant

Formulas for Small Matrices

Cofactor Expansion Method

Related pages

See also