Question

Find the nint of the element of second
and third in the determinat
3​

Answer 1

Basic Definition :-

Minor of a square matrix:-

Let us consider a square matrix A, then its minor is

$\sf \: represented \: by \: m_{ij} \: and \: is \: obtained \: by \: evaluating \: the$

$\sf \:determinant \: after \: deleting \: the \: {i}^{th} \: row \: and \: {j}^{th} \: column$

Given :-

$\rm :\longmapsto\:\begin{gathered}\sf A=\left[\begin{array}{ccc}3&-2&4\\5&2&1\\1&6& - 5\end{array}\right]\end{gathered}$

Now,

Minor of Second Column

$\rm :\longmapsto\:m_{12} \: = \: \begin{array}{|cc|}\sf 5 &\sf 1 \\ \sf 1 &\sf - 5 \\\end{array} = - 25 - 1 = - 26$

$\rm :\longmapsto\:m_{22} \: = \: \begin{array}{|cc|}\sf 3 &\sf 4 \\ \sf 1 &\sf - 5 \\\end{array} = - 15 - 4 = - 19$

$\rm :\longmapsto\:m_{23} \: = \: \begin{array}{|cc|}\sf 3 &\sf 4 \\ \sf 5 &\sf 1 \\\end{array} = 3 - 20 = - 17$

Minor of Third Column

$\rm :\longmapsto\:m_{13} \: = \: \begin{array}{|cc|}\sf 5 &\sf 2 \\ \sf 1 &\sf 6 \\\end{array} = 30 - 2 = 28$

$\rm :\longmapsto\:m_{23} \: = \: \begin{array}{|cc|}\sf 3 &\sf - 2 \\ \sf 1 &\sf 6 \\\end{array} = 18 + 2 = 20$

$\rm :\longmapsto\:m_{33} \: = \: \begin{array}{|cc|}\sf 3 &\sf - 2 \\ \sf 5 &\sf 2 \\\end{array} = 6 + 10 = 16$

Additional Information

Properties of Determinants :-

1. The determinant remains unaltered if its rows are changed into columns and the columns into rows.

2. If all the elements of a row (or column) are zero, then the determinant is zero.

3. If the all elements of a row (or column) are proportional (identical) to the elements of some other row (or column), then the determinant value is zero.

4. The interchange of any two rows (or columns) of the determinant changes its sign.

5. If all the elements of a determinant above or below the main diagonal consist of zeros, then the determinant is equal to the product of diagonal elements.