Determinant of a 3 X 3 Matrix – Definition, Formulas, Shortcut method and Examples

Definition of Determinant of a 3 x 3 Matrix

The determinant of a 3 x 3 matrix is a number that is calculated by taking the products of the matrix’s coefficients and subtracting the product of the matrix’s minor determinants. The determinant of a 3 x 3 matrix is also equal to the product of the determinants of the matrix’s three sub-matrices. Determinant of a 3 X 3 Matrix – Definition Formulas Shortcut method and Examples.

Importance of Determinant in Linear Transformation

A determinant is a function that takes a matrix as an input and outputs a scalar. The determinant is important in linear transformations because it helps to determine whether or not a transformation is invertible. If the determinant is zero, then the transformation is not invertible.

Finding Determinant of a 3×3 Matrix

The determinant of a 3×3 matrix is a number that is found by taking the products of the diagonal elements and then subtracting the products of the off-diagonal elements.

General Method

II

The second method of solving systems of linear equations is substitution. Substitution is a technique where one equation is solved for one variable and that variable is then substituted into the other equation. The variable is then solved for and the two solutions are compared. If they are the same, then the system is consistent and has a unique solution. If they are not the same, then the system is inconsistent and does not have a unique solution.

The substitution method is shown in the following example:

Example

Solve the system of equations using substitution.

2x + y = 5
3x – y = 2

Solution

The first equation can be solved for y.

y = 5 – 2x

The second equation can be solved for x.

x = 2y – 3

The two equations can be substituted into each other.

5 – 2x = 2y – 3
3x – 2y = 5

The two equations are consistent and have a unique solution.

Shortcut Method

The shortcut method is the fastest way to find the product of two numbers. To use the shortcut method, multiply the numbers in the numerator (top number) by the numbers in the denominator (bottom number).

For example, to find the product of 5 and 8, multiply 5 by 8 to get 40.

Uses of Determinant of a Matrix

There are a few primary uses of the determinant of a matrix. The determinant can be used to find the inverse of a matrix, to find the eigenvalues and eigenvectors of a matrix, and to calculate the volume of a parallelopiped formed by the vectors in a matrix. Additionally, the determinant can be used to solve systems of linear equations.

Determinant of a 3 X 3 Matrix – Definition Formulas Shortcut method and Examples.