Matrices and Determinants.

A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. The individual items in a matrix are called its elements or entries. Two matrices can be added or subtracted element by element if have the same number of rows and the same number of columns.

 

Order of matrix

The number of rows and columns that a matrix has is called its order or its dimension

Example $\left[ {\begin{array}{*{20}{c}}1&5&{ - 2}\\6&8&9\end{array}} \right]{\rm{\: }}$

The dimension of matrix is read as "two by three" because there are two rows and three columns.

Order or dimension = number of rows × number of columns

 

Components of matrix

Let M = $\left[ {\begin{array}{*{20}{c}}1&5&{ - 2}\\6&8&9\end{array}} \right]$

The numbers 1,5, -2,6,8,9 are the components of the matrix .It can be also written as

M = $\left[ {\begin{array}{*{20}{c}}{{{\rm{a}}_{11}}}&{{{\rm{a}}_{12}}}&{{{\rm{a}}_{13}}}\\

{{{\rm{a}}_{21}}}&{{{\rm{a}}_{22}}}&{{{\rm{a}}_{23}}}\end{array}} \right]$

Where a₁₁ denotes first row first column a₁₁ = 1 similarly a₁₂ = 5 and so on.

 

Types of matrix

 

Row matrix

Matrix having only one row is called row matrix. The order of such matrix is of 1× n

Example M = [5, 6,8] is the matrix of order 1× 3

 

Column matrix

Matrix containing only one column is called column matrix. The order of such matrix is of n ×1

Example M = $\left( {\begin{array}{*{20}{c}}5\\6\\{ - 9}\end{array}} \right){\rm{\: }}$is the matrix of order 1× 3

 

Null matrix or Zero matrix

Matrix having all elements zero is called zero or null matrix.

M = ${\left( {\begin{array}{*{20}{c}}0\\0\\0\end{array}} \right)_{3* 1}}$

 

M = ${\left[ {\begin{array}{*{20}{c}}0&0&0\\0&0&0\end{array}} \right]_{2* 3}}$

 

 

Square matrix

 

A square matrix is a matrix with an equal number of rows and columns.

Example: M = ${\left( {\begin{array}{*{20}{c}}5&6&7\\6&9&{ - 6}\\{ - 44}&8&{ - 99}

\end{array}} \right)_{3* 3}}$

M is a square matrix of order 3 × 3

 

Diagonal matrix 

A diagonal matrix is a square matrix that has all its elements zero except for those in the diagonal from top left to bottom right.

Example: M = ${\left( {\begin{array}{*{20}{c}}5&0&0\\0&9&0\\0&0&{ - 9}\end{array}} \right)_{3* 3}}$

Scalar matrix

A diagonal matrix having same main diagonal element is called scalar matrix

Example: M = $\left( {\begin{array}{*{20}{c}}5&0&0\\0&5&0\\0&0&5\end{array}} \right)$

Unit matrix

A unit matrix is a diagonal matrix whose elements in the diagonal are all ones.

 I = ${\left( {\begin{array}{*{20}{c}}1&0\\{{\rm{\: }}0}&1\end{array}} \right)_{2* 2}}$

 

I = ${\left( {\begin{array}{*{20}{c}}1&0&0\\0&1&0\\0&0&1\end{array}} \right)_{3* 3}}$

I is a unit matrix.

Symmetric matrix:

A square matrix which do not change its row and column are interchanged is called symmetric matrix.

S = ${\left( {\begin{array}{*{20}{c}}2&5&6\\5&8&4\\6&4&9\end{array}} \right)_{3* 3}}$

 

 

Transpose of matrix

The transpose of a matrix is a matrix which is obtained by interchanging rows into column and column into rows. It is denoted by A’ or A^T

Example 1

 M = $\left[ {\begin{array}{*{20}{c}}1&5&{ - 2}\\6&8&9\end{array}} \right]$ transpose of M = T

 T = $\left( {\begin{array}{*{20}{c}}1&6\\5&8\\{ - 2}&9\end{array}} \right)$

 

Multiplication of matrices

Rules:

1. The number of columns in the 1st one equals the number of rows in the 2nd one

2. Multiply the elements of each row of the first matrix by the elements of each column in the second matrix.

3. Add the products.

Example 2

 

 $\left( {\begin{array}{*{20}{c}}{\rm{a}}&{\rm{x}}\end{array}} \right)\left( \begin{array}{*{20}{c}}

{\rm{p}}&2\\3&{\rm{q}}\end{array}} \right)$ = ( a.p + 3x , 2a + xq )

 

Properties of the matrices multiplication

Associative property: A (BC) = A (BC)

Distributive property: A (B +C) = AB + BC

Identity property: AI = A = IA

 

 

Singular and non – singular matrices

 

Singular matrix is square matrix whose determinant is equal to Zero 

Example

 A = $\left( {\begin{array}{*{20}{c}}5&5\\2&2\end{array}} \right)$ is the singular matrix

|A| = 10 -10 = 0

 

Non - Singular matrix is also square matrix whose determinant is not equal to zero.

A = $\left( {\begin{array}{*{20}{c}}6&5\\2&2\end{array}} \right)$ is the non singular matrix

|A| = 12 - 10 = 2

 

Determinant of 2 × 2 matrix

 

If A = $\left( {\begin{array}{*{20}{c}}{\rm{a}}&{\rm{b}}\\{\rm{c}}&{\rm{d}}\end{array}} \right)$ be 2 * 2 then its determinant

|A |= ad -bc                                            

 

Inverse matrix

For a matrix A its inverse B exist when AB = BA = I exists.

A = $\left( {{\rm{\: }}\begin{array}{*{20}{c}}{\rm{a}}&{\rm{b}}\\{\rm{c}}&{\rm{d}}\end{array}{\rm{\: }}} \right)$ B=$\left({{\rm{\:}}\begin{array}{*{20}{c}}{\rm{e}}&{\rm{f}}\\{\rm{g}}&{\rm{h}}\end{array}{\rm{\: }}} \right)$

 

The components of the inverse matrix can be obtained by

e =${\rm{\: }}\frac{{\rm{d}}}{{{\rm{ad}} - {\rm{bc\: }}}}$

g = $ - \frac{{\rm{c}}}{{{\rm{ad}} - {\rm{bc\: }}}}$

f = $ - \frac{{\rm{b}}}{{{\rm{ad}} - {\rm{bc}}}}$

h = $\frac{{\rm{a}}}{{{\rm{ad}} - {\rm{bc}}}}$

 

We have matrix equation

 AX = B

If A^-1 exists if, |A| ≠ o

⇒ A^-1 (AX) = A^-1 B

⇒I.X = A^-1 B

⇒ X= A^-1 B

By using this equation we can solve simultaneous equations

 

Example 3

Find the value of x and y

 

$\left( {\begin{array}{*{20}{c}}5&{\rm{x}}\\{\rm{y}}&7\end{array}} \right)\left( {\begin{array}{*{20}{c}}1\\{ - 2}\end{array}} \right) = {\rm{\: }}\left( {\begin{array}{*{20}{c}}{19}\\{ - 4}\end{array}} \right)$

$\left( {\begin{array}{*{20}{c}}{5 - 2{\rm{x\: }}}\\{{\rm{y}} - 14{\rm{\: }}}\end{array}{\rm{\: }}} \right)$ = $\left( {\begin{array}{*{20}{c}}{19}\\{ - 4}s\end{array}} \right)$

Now,

 5-2x = 19

∴ x = -7

And

y-14 = -4

∴ y = 10