Question

How do you Evaluate Determinants?

Answer 1

Hint : The determinant can be expressed as the scalar value which can be computed from the elements of a square matrix and encodes certain properties of the linear transformation. The determinant is denoted by det(A), det A, or |A|.

Complete answer:
In linear and the multilinear algebra, the determinant is associated with the element of a square matrix with “n” number of rows and the “n” number of columns.
The order of the matrix is denoted by $2 \times 2$ with two rows and two columns and similarly it is denoted by $3 \times 3$ for three rows and three columns.
Now, the process of evaluating is quite messy and therefore let’s start with the order $2$ matrix, that is $2 \times 2$ matrix.
Now, take the determinant of the order $2 \times 2$ , in which you have to multiply the top left member with the bottom right diagonal and have to subtract the product of the bottom left with the top right member of the diagonal.
For Example:
\left| {\begin{array}{*{20}{c}} a&b; \\\ c&d; \end{array}} \right|
Expand the above determinant which gives: $ad - bc$
Similarly, for the order $3$ determinant that is $3 \times 3$
For Example: \left| {\begin{array}{*{20}{c}} a&b;&c; \\\ d&e;&f; \\\ g&h;&i; \end{array}} \right|
To expand the determinant, you have to find the minor determinants and the product of the element of the first rows one by one.
For the minor determinant for the first row and first column element you remove then and note down the element for $2 \times 2$ matrix. Similarly, for the second element and the third element of the first row. Also, be careful about the sign convention like it will be plus, minus and then plus.
Expansion of the determinant gives:
= a\left| {\begin{array}{*{20}{c}} e&f; \\\ h&i; \end{array}} \right| - b\left| {\begin{array}{*{20}{c}} d&f; \\\ g&i; \end{array}} \right| + c\left| {\begin{array}{*{20}{c}} d&e; \\\ g&h; \end{array}} \right|
Again, simplify the minor determinants.
$= a(ei - fh) - b(di - fg) + c(dh - eg)$

Note : Don’t be confused between the determinants and the matrices. Determinant is the square matrix with the same number of rows and columns whereas, the matrix is the rectangular grid of numbers and number of rows and the columns may not be the same.