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Determinant

From Simple English Wikipedia, the free encyclopedia

Thedeterminant of asquare matrix is ascalar (a number) that indicates how thatmatrix behaves. It can be calculated from the numbers in the matrix.

The determinant of the matrix $A {\displaystyle A}$ is written as $\det(A)$ or $|A|$ in a formula.^[1]^[2] 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 $\left|{\begin{matrix}a&b\\c&d\end{matrix}}\right|$ .

Interpretation

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There are a few ways to understand what the determinant says about a matrix.

Geometric interpretation

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For a $2\times 2$ matrix ${\begin{bmatrix}a&c\\b&d\end{bmatrix}}$ , the determinant is the area of a parallellogram. (The area is equal to $ad-bc$ .)

{\displaystyle 2\times 2} — For a $2\times 2$ matrix ${\begin{bmatrix}a&c\\b&d\end{bmatrix}}$ , the determinant is the area of a parallellogram. (The area is equal to $ad-bc$ .)

An $n\times n$ matrix can be seen as describing alinear map in $n {\displaystyle n}$ dimensions. In which case, the determinant indicates thefactor by which this matrixscales (grows or shrinks) a region of $n {\displaystyle n}$ -dimensional space.

For example, a $2\times 2$ matrix $A {\displaystyle A}$ , seen as a linear map, will turn a square in 2-dimensional space into aparallelogram. That parallellogram's area will be $\det(A)$ times as big as the square's area.

In the same way, a $3\times 3$ matrix $B {\displaystyle B}$ , seen as a linear map, will turn acube in 3-dimensional space into aparallelepiped. That parallelepiped's volume will be $\det(B)$ times as big as the cube's volume.

The determinant can be negative or zero. A linear map can stretch and scale a volume, but it can also reflect it over anaxis. Whenever this happens, thesign of the determinant changes from positive to negative, or from negative to positive. A negative determinant means that the volume was mirrored over anodd number of axes.

"System of equations" interpretation

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One can think of a matrix as describing asystem of linear equations. That system has a unique non-trivial solutionexactly when the determinant is not 0^[2] (non-trivial meaning that the solution is not just all zeros).

If the determinant is zero, then there is either no unique non-trivial solution, or there areinfinitely many.

Singular matrices

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A matrix has aninverse matrix exactly when the determinant is not 0. For this reason, a matrix with a non-zero determinant is calledinvertible. If the determinant is 0, then the matrix is callednon-invertible orsingular.^[2]

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

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There are a few ways to calculate a determinant.

Formulas for small matrices

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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.

{\displaystyle 3\times 3} — 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:^[2]

\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:^[3]

{\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 theRule of Sarrus (see image) to remember this formula.

Cofactor expansion

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

Suppose that we have an $n\times n$ matrix $A {\displaystyle A}$ . First, we choose any row or column of the matrix. For each number $a_{ij}$ in that row or column, we calculate something called itscofactor $C_{ij}$ . Then $\det(A)=\sum a_{ij}C_{ij}$ .^[2]

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

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

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.

Related pages

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References

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↑"Comprehensive List of Algebra Symbols".Math Vault. 2020-03-25. Retrieved2020-09-09.
↑^2.0^2.1^2.2^2.3^2.4Weisstein, Eric W."Determinant".mathworld.wolfram.com. Retrieved2020-09-09.
↑"Determinant of a Matrix".www.mathsisfun.com. Retrieved2020-09-09.

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