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- Square Matrix: A matrix whose number of columns is equivalent to its number of rows. (n x n) | - Square Matrix: A matrix whose number of columns is equivalent to its number of rows. (n x n) | ||

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## Revision as of 08:25, 9 December 2012

## Contents

## THE DETERMINANT OF A MATRIX

## Definitions

- Square Matrix: A matrix whose number of columns is equivalent to its number of rows. (n x n)

- Determinant:

## How to Find the Determinant of...

**2 x 2 Matrices**

A = $ \left[\begin{array}{cccc}a&b\\c&d\end{array}\right] $

**3 x 3 Matrices**

A = $ \left[\begin{array}{cccc}a&b&c\\d&e&f\\g&h&i\end{array}\right] $

**4 x 4 Matrices**

A = $ \left[\begin{array}{cccc}a&b&c&d\\e&f&g&h\\i&j&k&l\\m&n&o&p\end{array}\right] $

## Examples

**2 x 2 Matrix**

A = $ \left[\begin{array}{cccc}4&2\\-3&1\end{array}\right] $

det(A) = (4)*(1) - (2)*(-3)

= 4-(-6) = 10

**3 x 3 Matrix**

A = $ \left[\begin{array}{cccc}2&5&1\\3&2&2\\1&5&2\end{array}\right] $

det(A) = 2*[(2*2) - (2*5)] - 5*[(3*2) - (2*1)] + 1*[(3*5) - (2*1)]

= 2*(4 - 10) - 5*(6 - 2) + 1*(15 - 2) = 2*(-6) - 5*(4) + 1*(13) = -12 - 20 + 13 = -19

**4 x 4 Matrix**

A = $ \left[\begin{array}{cccc}2&3&7&2\\1&6&2&3\\1&1&4&3\\4&6&5&1\end{array}\right] $

det(A) = 2*{6*[(4*1) - (3*5)] - 2*[(1*1) - (3*6)] + 3*[(1*5) - (4*6)]} - 3*{1*[(4*1) - (3*5)] - 2*[(1*1) - (3*4)] + 3*[(1*5) - (4*4)]} + 7*{1*[(1*1) - (3*6)] - 6*[(1*1) - (3*4)] + 3*[(1*6) - (1*4)]} - 2*{1*[(1*5) - (4*6)] - 6*[(1*5) - (4*4)] + 2*[(1*6) - (1*4)]}

= 2*[6*(4-15) - 2*(1-18) + 3*(5-24)] - 3*[1*(4-15) - 2*(1-12) + 3*(5-16)] + 7*[1*(1-18) - 6*(1-12) + 3*(6-4)] - 2*[1*(5-24) - 6*(5-16) + 2*(6-4)] = 2*[(6*(-11)) - (2*(-17)) + (3*(-19))] - 3*[(1*(-11)) - (2*(-11)) + (3*(-11))] + 7*[(1*(-17)) - (6*(-11)) + (3*2)] - 2*[(1*(-19)) - (6*(-11)) + (2*2)] = 2*(-66 - (-34) + (-57)) - 3*(-11 - (-22) + (-33)) + 7*((-17) - (-66) + 6) - 2*((-19) - (-66) + 4) = 2*(-89) - 3*(-22) + 7*(55) - 2*(51) = -178 - (-66) + 385 - 102 = 171

## Important Properties of the Determinant

- The determinant of the transpose of a matrix will be the same as the determinant of the original matrix.

- If two rows in the matrix are interchanged, the determinant will be the negative of the determinant of the original matrix.

- If two rows or columns in a matrix are the same, the determinant will be zero.

- If there is a row of zeros in the matrix, the determinant will be zero.

- If a row or column in a matrix was multiplied by a real number, then the determinant will be that real number multiplied by the determinant of the original matrix.

- If the matrix is either an upper or a lower triangular matrix, the determinant will be the product of the elements on the main diagonal.

- If two matrices are n x n, then the determinant of one multiplied by the determinant of the other is the determinant of the full matrix of the two next to each other.

## Nonsingular and Similar Matrices

Definition: A square matrix is called **nonsingular** if its determinant is not equal to zero.

- For square matrices, then the linear combination of the matrix has a nontrivial solution if and only if the matrix is NOT nonsingular.

- For nonsingular matrices, the determinant of the inverse of that matrix is equal to 1 over the determinant of the matrix.

- If two matrices are similar matrices, then the determinants of the two are equal.