Linear Algebra - Determinant

Determinant

Property of Determinant

DETERMINANT PROPERTY 1: If every element in a row or column of a matrix is 0, then the determinant equals 0.

$$\det \left(\left|\begin{array}{cc}2& 4\\ 0& 0\end{array}\right|\right)=0$$

DETERMINANT PROPERTY 2: If corresponding rows and columns of a matrix are interchanged, the determinant is not changed.

$$\det \left(\left|\begin{array}{cc}1& 3\\ 5& 7\end{array}\right|\right)=\det \left(\left|\begin{array}{cc}1& 5\\ 3& 7\end{array}\right|\right)=1\times 7-3\times 5=-8$$

so, by property 2, we have $\det (A)=\det (B)=\det (A^T)$

$$A=\left[\begin{array}{rrr}2& 1& 6\\ 3& 0& 5\\ -4& 6& 9\end{array}\right]\text{ and }B=\left[\begin{array}{rrr}2& 3& -4\\ 1& 0& 6\\ 6& 5& 9\end{array}\right]$$

DETERMINANT PROPERTY 3: Interchanging two rows (or columns) of a matrix reverses the sign of the determinant.
DETERMINANT PROPERTY 4: If every element of a row (or column) of a matrix is multiplied by the real number k, then the determinant of the new matrix is k times the determinant of the original matrix.
DETERMINANT PROPERTY 5:  The determinant of a matrix with two identical rows (or columns) equals 0.
DETERMINANT PROPERTY 6: The determinant of a matrix is unchanged if a multiple of a row (or column) of the matrix is added to the corresponding elements of another row (or column).
DETERMINANT PROPERTY 7： The determinant of a triangular matrix is a product of its principal diagonal elements.

$$A=\left(\begin{array}{rrrr}4& -2& 6& 2\\ 0& -1& 5& -3\\ 2& -1& 8& -2\\ 0& 0& 0& 2\end{array}\right)$$

Subtracting the first row, divided by 2, from the third row, we get

$$\begin{array}{rl}& \det A=\left(\begin{array}{rrrr}4& -2& 6& 2\\ 0& -1& 5& -3\\ 2& -1& 8& -2\\ 0& 0& 0& 2\end{array}\right)=2\cdot \det \left(\begin{array}{rrrr}2& -1& 3& 1\\ 0& -1& 5& -3\\ 2& -1& 8& -2\\ 0& 0& 0& 2\end{array}\right)=2\cdot \det \left(\begin{array}{rrrr}2& -1& 3& 1\\ 0& -1& 5& -3\\ 0& 0& 5& -3\\ 0& 0& 0& 2\end{array}\right)=2\cdot 2\cdot (-1)\cdot 5\cdot 2=-40\end{array}$$

Here, we make it as upper triangular matrix, then use the last property, multiple all the diagonal elements

DETERMINANT PROPERTY 8： Two columns(rows) are proptional to each other, then the det is 0.

Example:

Without expanding, show that the determinant of the following matrix is zero.

$$\left[\begin{array}{rrr}2& 5& -1\\ 1& -15& 3\\ -2& 10& -2\end{array}\right]$$

Each element in the second column equals -5 times the corresponding element in the third column. Multiply the elements of the third column by 5 and then add the results to the corresponding elements in the second column to get a matrix with an equivalent determinant,

$$\left[\begin{array}{rrr}2& 0& -1\\ 1& 0& 3\\ -2& 0& -2\end{array}\right]$$

By Property 1, the determinant of this matrix is zero.

Therefore, if two columns or two rows are proptional to each other, then the det is 0, can be simply proved by property 6 + 1