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Determinants of matrices

The determinant is a real function  such that each square matrix \mathbf{A} joins a real number (the determinant of a matrix \mathbf{A}). The determinant of a square matrix \mathbf{A} is denoted as det \mathbf{A} or |\mathbf{A}|.

The determinant of matrices we define as inductive, that is, the determinant of a square matrix of the n-th order we define using the determinant of a square matrix of the (n-1) -th order.

The determinant of a matrix \mathbf{A}=[a] of order 1 is the number a:

    \[det \mathbf{A}=a.\]

The determinant of a matrix \mathbf{A} of order 2, \mathbf{A}=\left[ \begin {array} {cc}  a & b \\  c & d \\  \end{array} \right] , is the number:

    \[det \mathbf{A} = ad-bc\]

.

The determinant of a matrix \mathbf{A} of order 3,

\mathbf{A}=\left[ \begin {array} {ccc}  a & b & c \\  d & e & f \\  g & h & i \\  \end{array} \right] , is the number:

    \[det \mathbf{A} = a \left| \begin {array} {cc} e & f \\ h & i \\ \end{array} \right| - d \left| \begin {array} {cc} b & c \\ h & i \\ \end{array} \right| + g \left | \begin {array} {cc} b & c \\ e & f \\ \end{array} \right|.\]

    \[\vdots\]

The determinant of a matrix

    \[\mathbf{A} = \left[ \begin{array} {ccccc} a_{11} & a_{12} & a_{13} & \ldots & a_{1n} \\ a_{21} & a_{22} & a_{23} & \ldots & a_{2n} \\ a_{31} & a_{32} & a_{33} & \ldots & a_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots\\ a_{n1} & a_{n2} & a_{n3} & \ldots & a_{nn} \\ \end{array} \right]\]

of the n-th order is the number

    \[det \mathbf{A} = a_{11} det \mathbf{A}_{11} + a_{21} det \mathbf{A}_{21} + \ldots + (-1) ^{n-1} a_{n1} det \mathbf{A}_{n1}.\]

Example 1.

Calculate the determinant of the following matrices:

a)  \mathbf{A}= \left[ \begin {array} {cc}  -1 & 5 \\  3 & -2 \\  \end{array} \right]

b) \mathbf{B}= \left[ \begin {array} {ccc}  2 & 1 & -2 \\  3 & 4 & -6 \\  -3 & 0 & 9 \\  \end{array} \right].

Solution:

a)

    \[\left| \begin {array} {cc} -1 & 5 \\ 3 & -2 \\ \end{array} \right| = (-1) \cdot (-2) - 5 \cdot 3 = 2 - 15 = -13\]

b)

    \[\left| \begin {array} {ccc} 2 & 1 & -2 \\ 3 & 4 & -6 \\ -3 & 0& 9 \\ \end{array} \right| = 2 \left| \begin {array} {cc} 4 & -6 \\ -3 & 9 \\ \end{array} \right| - 3 \left| \begin {array} {cc} 1 & -2 \\ 0 & -9 \\ \end{array} \right| + (-3) \left| \begin {array} {cc} 1 & -2 \\ 4 & -6 \\ \end{array} \right|=\]

    \[= 2 \cdot (4\cdot 9 - (-6) \cdot (-3)) + 3\cdot  (1 \cdot 9 - (-2) \cdot 0) -3 \cdot (1 \cdot (-6) - (-2) \cdot 4) =\]

    \[2 \cdot  (36-18) + 3 \cdot (9 + 0) - 3 \cdot (-6 +8) =\]

    \[2 \cdot 18 + 3 \cdot 9 - 3 \cdot 2 = 36 + 27 -6 = 57\]

As we can see, calculating the determinant it is not easy for matrices of a higher order. There are several methods for calculating the determinant. The general method is the Laplace expansion of a determinant along a given column or row.

Let \mathbf{A}=[a_{ij}] be a square matrix of order n. If in a matrix \mathbf{A} is removed i-th row and j-th column, we have thus obtained a matrix of order (n-1) of  which the determinant is called a minor and is denoted as M_{ij}.

The algebraic complement or cofactor of an element a_{ij} is the number:

    \[A_{ij}=(-1)^{i+j}M_{ij}.\]

Let \mathbf{A} be an n\times n matrix. For a matrix \mathbf{A} we have the following expansions:

a) for a fixed i :

    \[det \mathbf{A}= \sum_{j=1}^n (-1)^{i+j} a_{ij} M_{ij},\]

b) for a fixed j

    \[det \mathbf{A}= \sum_{i=1}^n (-1)^{i+j} a_{ij} M_{ij}.\]

 

The properties of determinants

 

Let \mathbf{A} be a matrix of order n.

1.) A matrix \mathbf{A} and its transpose matrix \mathbf{A}^T have the same determinants, that is

    \[det \mathbf{A} = det \mathbf{A}^T.\]

2.) If \mathbf{A} is a triangular matrix, then its determinant is equal to the product of all diagonal elements, that is

    \[det \mathbf{A} = a_{11} \cdot a_{22}\cdot \ldots \cdot a_{nn}.\]

3.) If two columns or rows of the determinant interchange the location, then

    \[det \mathbf{A}= - det \mathbf{A}.\]

4.) If a matrix \mathbf{A} has two equal columns or rows, then det \mathbf{A}=0.

5.) If \mathbf{A} is a matrix with a row or column where every element is equal to zero, then det \mathbf{A}=0.

6.) The determinant value does not change if to one column we add a linear combination of the remaining columns.

7.) If a matrix \mathbf{B} is obtained from a matrix \mathbf{A} by multiplying one its row (or column) with a scalar \alpha, then

    \[det \mathbf{B} = \alpha \cdot det \mathbf{A}.\]

8.) A matrix \mathbf{A} is a regular matrix iff det \mathbf{A} \neq 0.

9.) For a singular matrix \mathbf{A} is det\mathbf{A} =0.

Example 2.

Using the Laplace expansion of a determinant along 4-th column, calculate the determinant of the following matrix:

    \[\mathbf{A} = \left[ \begin {array} {cccc} 1 & -4 & -2 & 0 \\ 3 &  5 &  0 &-1 \\ 7 & 0& -9 & 0 \\ -2 & 1& -3 & 2 \\ \end{array} \right].\]

Solution:

    \[\left| \begin {array} {cccc} 1 & -4 & -2 & 0 \\ 3 &  5 &  0 &-1 \\ 7 & 0& -9 & 0 \\ -2 & 1& -3 & 2 \\ \end{array} \right| = (-1)^{2+4} \cdot (-1) \cdot \left| \begin {array} {ccc} 1 & -4 & -2  \\ 7 & 0& -9  \\ -2 & 1& -3  \\ \end{array} \right| + 2 \cdot \left| \begin {array} {ccc} 1 & -4 & -2  \\ 3 & 5 & 0  \\ -2 & 1& -3  \\ \end{array} \right| =\]

    \[=1 \cdot (-1) \cdot \left( 1 \cdot \left| \begin {array} {cc} 0 & -9 \\ 1 & -3   \\ \end{array} \right| - 7 \cdot \left| \begin {array} {cc} -4 & -2 \\ 1 & -3 \\ \end{array} \right| + (-2) \cdot \left| \begin {array} {cc} -4 & -2 \\ 0 & -9 \\ \end{array} \right| \right) +\]

    \[+2 \cdot \left( 1 \cdot \left| \begin {array} {cc} 5 & 0 \\ 1 & -3   \\ \end{array} \right| - 3 \cdot \left| \begin {array} {cc} -4 & -2 \\ 1 & -3 \\ \end{array} \right| + (-2) \cdot \left| \begin {array} {cc} -4 & -2 \\ 5 & 0   \\ \end{array} \right| \right)  \\ =\]

    \[= (-1) \cdot (-161) + 2 \cdot (-47) = 161 - 94 = 67.\]

In general, we choose the row or column that contains the most zeroes, because then we have a shorter calculation.

 

The Binet – Cauchy theorem.

 

If matrices \mathbf{A} and \mathbf{B} are square matrices of the same order, then

    \[\det (\mathbf{A}\mathbf{B}) = det \mathbf{A} det \mathbf{B}\]

is valid.

We can observe that for a regular matrix \mathbf{A}

    \[\mathbf{I} = \mathbf{A} \mathbf{A}^{-1}\]

is valid if we know the determinant of the matrix \mathbf{A} and through using the Binet – Cauchy theorem we can calculate \mathbf{A}^{-1}. Namely,

    \[\mathbf{A}^{-1}= \frac{1}{det \mathbf{A}}.\]

 

The adjugate or adjoint of a square matrix \mathbf{A} is a matrix \mathbf{\tilde A} = [x_{ij}], where x_{ij} = a_{ji}, \forall i, j = 1, \ldots, n. This means that the adjugate of a square matrix \mathbf{A} is the transpose of a cofactor matrix of a matrix \mathbf{A}.

The adjugate of a matrix \mathbf{A} can also be denoted as adj (\mathbf{A}).

For instance, the adjugate matrix of a matrix \mathbf{A} =\left[ \begin {array} {cc}  1 & -2 \\  3 & 4   \\  \end{array} \right] is a matrix \mathbf{\tilde A} =\left[ \begin {array} {cc}  4 & -3 \\  2 & 1   \\  \end{array} \right].

As was mentioned previously, a square matrix \mathbf{A} is a regular matrix iff det \mathbf{A}  \neq 0. In this case, an inverse matrix \mathbf{A}^{-1} of a matrix \mathbf{A} is given by the following formula:

    \[A^{-1} = \frac{1}{det \mathbf{A}} \mathbf{\tilde A}.\]

Now we can find, using the formula above, an inverse matrix of a given square matrix of any order.

 

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