# Matrices

An introduction to matrices

A square matrix is a matrix with the same number of rows as columns.

A matrix with m rows and n columns is an m x n matrix. This is called the order.

You can only add or subtract matrices with the same order.

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## Multiplying

To multiply matrices, the number of columns in the first one must be the same as the number of rows in the second one, otherwise they do not conform and cannot be multiplied.

Remember to use each row of the first matrix with each column of the second.

Matrix multiplication is associative - P(QR) = (PQ)R

Matrix multiplication is not commutative - AB is not the same as BA

Matrix multiplication is distributive - P(Q+R) = PQ + PR

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## Identity Matrix

This is denoted by I

This has ones down the main diagonal and 0 in all the other positions.

When you multiply any matrix A by I you get A as the answer

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## Zero matrix

This exists when all the entries are 0.

A diagonal matrix has zeros everywhere except the leading diagonal.

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## Transpose

To find the transpose of a matrix you swap the rows and columns over.

The transpose of A is written as A^T

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## Determinant

The determinant of the 2x2 matrix (abcd) is defined as the number ad-bc.

A matrix is called singular if the detereminant is 0

¦A¦ is another way of writing det(A)

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## Inverse

If AB = BA = I , then A is the inverse of B (and likewise B is the inverse of A) therefore A = B^-1 and B=A^-1

To find the inverse of a 2x2 matrix, swap the elements on the leading diagonal and change the sign of the other two. Then divide the new matrix my the determinant of the original matrix.

When the determinant is 0, the inverse matrix does not exist so we can only  find inverses for non-singular matrices.

The inverse of AB = B^-1A^-1

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## Simultaneous equations

These can be solved using 2x2 matrices.

If det=0 the  there is no unique solution and we have toue a parameter y=t to solve it. We say that the equations are inconsistent.

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