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Determinant facts for kids

Kids Encyclopedia Facts

The determinant of a square matrix is a scalar (a number) that indicates how that matrix behaves. It can be calculated from the numbers in the matrix.

The determinant of the matrix A is written as \det(A) or |A| in a formula. Sometimes, instead of \det\left(\begin{bmatrix}a&b\\c&d\end{bmatrix}\right) and \left|\begin{bmatrix}a&b\\c&d\end{bmatrix}\right|, one simply writes \det\begin{bmatrix}a&b\\c&d\end{bmatrix} and \left|\begin{matrix}a&b\\c&d\end{matrix}\right|.

Singular matrices

A matrix has an inverse matrix exactly when the determinant is not 0. For this reason, a matrix with a non-zero determinant is called invertible. If the determinant is 0, then the matrix is called non-invertible or singular.

Geometrically, one can think of a singular matrix as "flattening" the parallelepiped into a parallelogram, or a parallelogram into a line. Then the volume or area is 0, which means that there is no linear map that will bring the old shape back.

Calculating a determinant

There are a few ways to calculate a determinant.

Formulas for small matrices

Sarrus rule vertical
The 3 \times 3 determinant formula is a sum of products. Those products go along diagonals that "wrap around" to the top of the matrix. This trick is called the Rule of Sarrus.
  • For 1 \times 1 and 2 \times 2 matrices, the following simple formulas hold:

    \det\begin{bmatrix}a\end{bmatrix} = a,\qquad\det\begin{bmatrix}a&b\\c&d\end{bmatrix} = ad-bc.

  • For 3 \times 3 matrices, the formula is:

    {\det\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix} = {\color{blue}{aei}+{dhc}+{gbf}}{\color{OrangeRed}{}-{gec}-{ahf}-{dbi}}}

    One can use the Rule of Sarrus (see image) to remember this formula.

Cofactor expansion

For larger matrices, the determinant is harder to calculate. One way to do it is called cofactor expansion.

Suppose that we have an n \times n matrix A. First, we choose any row or column of the matrix. For each number a_{ij} in that row or column, we calculate something called its cofactor C_{ij}. Then \det(A) = \sum a_{ij} C_{ij}.

To compute such a cofactor C_{ij}, we erase row i and column j from the matrix A. This gives us a smaller (n-1)\times(n-1) matrix. We call it M. The cofactor C_{ij} then equals (-1)^{i+j} \det(M).

Here is an example of a cofactor expansion of the left column of a 3 \times 3 matrix:

\begin{align}
\det \begin{bmatrix}{\color{red}1}&3&2\\{\color{red}2}&1&1\\{\color{red}0}&3&4\end{bmatrix}
&= {\color{red}1} \cdot C_{11} + {\color{red}2} \cdot C_{21} + {\color{red}0} \cdot C_{31}
\\ &= \left( {\color{red}1} \cdot (-1)^{1+1} \det\begin{bmatrix}1&1\\3&4\end{bmatrix} \right)
+ \left( {\color{red}2} \cdot (-1)^{2+1} \det\begin{bmatrix}3&2\\3&4\end{bmatrix} \right)
+ \left( {\color{red}0} \cdot (-1)^{3+1} \det\begin{bmatrix}3&2\\1&1\end{bmatrix} \right)
\\ &= ({\color{red}1} \cdot 1 \cdot 1) + ({\color{red}2} \cdot (-1) \cdot 6) + {\color{red}0}
\\ &= -11.
\end{align}

As illustrated above, one can simplify the computation of determinant by choosing a row or column that has many zeros; if a_{ij} is 0, then one can skip calculating C_{ij} altogether.

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