properties of determinant proves

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properties of determinant proves

Properties of determinant proves:

property of determinant number 1:

If the rows and columns of a determinant are changed, the value of the determinant does not change.

properties of determinant proves

EXAMPLE:

determinant property 1 analytic proof
Row and column interchanged, the result is the same.

 

Property of determinant number 2:

The value of the determinant changed sign if any two rows or column interchanged

EXAMPLE:

\small \begin{vmatrix} a_{11} &a_{12} \\ a_{21}&a_{22} \end{vmatrix}=a_{11}a_{22}-a_{12}a_{21}\: \: and\, when\: \: column\, \, interchanged

\small \small \begin{vmatrix} a_{12} &a_{11} \\ a_{22}&a_{21} \end{vmatrix}=a_{12}a_{21}-a_{11}a_{22}=-\left ( a_{11}a_{22}-a_{12}a_{21} \right ) =-\small \begin{vmatrix} a_{11} &a_{12} \\ a_{21}&a_{22} \end{vmatrix}
When column interchanged with negative sign, same result

property of determinant number 3:

If all the entries in any row or column are zero, the value of the determinant is zero.

EXAMPLE:

Properties of determinant proves

determinant of any row or column is zero the result is zero
determinant of any row or column is zero, the result is zero

property of determinant number 4:

If any two rows or columns of a determinant are identical , the value of the determinant is zero.

EXAMPLE:

\small \begin{vmatrix} a & b &c \\ a & b &c \\ x & y & z \end{vmatrix}=0

Two rows or columns are identical proofs

\small \begin{vmatrix} 1 &2 & 3\\ 1 &2 &3 \\ 4 &5 &6 \end{vmatrix}=1(15-12)-2\left (6- 12 \right )+3\left ( 5-8 \right )=0

property of determinant number 5:

If any two row or column of a determinant is multiplied by a non-zero number, k

the value of the new determinant becomes equal to K times the value of the original determinant.

EXAMPLE:

\small \left | A \right |=\begin{vmatrix} a_{11} &a_{12} \\ a_{21}&a_{22} \end{vmatrix}    multiplying first row by a non-zero K, we get

\small \begin{vmatrix} ka_{11} &ka_{12} \\ a_{21}&a_{22} \end{vmatrix}=ka_{11}a_{22}-ka_{12}a_{21}=k\left ( a_{11}a_{22}-a_{12}a_{21} \right )=k\begin{vmatrix} a_{11} &a_{12} \\ a_{21}&a_{22} \end{vmatrix}

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