12th Class Mathematics Matrices Notes - Mathematics Olympiads -Matrices and Determinats

 

                                                                                       Matrices and Determinants

 

• In previous classes, we have learnt the methods to solve linear equations. Let us consider the linear equations

\[{{a}_{1}}x+{{b}_{1}}y={{c}_{1}}\]                        (i)

\[{{a}_{2}}x+{{b}_{2}}y={{c}_{2}}\]                        (ii)

We have one of the methods to solve these equation by cross multiplication method.

\[\frac{x}{{{b}_{1}}{{c}_{2}}-{{b}_{2}}{{c}_{1}}}=\frac{y}{{{c}_{1}}{{a}_{2}}-{{c}_{2}}{{a}_{1}}}=\frac{1}{{{a}_{1}}{{b}_{2}}-{{a}_{2}}{{b}_{1}}}\]

Now, we learn to solve such equations with the help of matrices and determinants.

 

• Matrix: A matrix is a rectangular array of numbers or expressions arranged in rows and columns. It is the shorthand of mathematics. It is an operator as addition, multiplication etc. Every matrix has come into existence through the solution of linear equations.

Given linear equation can be solved by matrix method & it is written as,

AX = B

Where

 

 

• Definition: It is the arrangement of things into horizontal rows and vertical column. Generally matrix is represented by [ ] (square bracket) or ( ) etc.

Generally, it is represented as, where \[A=[{{a}_{ij}}]\]     

i = 1, 2, 3, ........ m

j = 1, 2, 3, ....... n.

Here subscript i denotes no. of rows.

& subscript j determines no. of columns.

& \[{{a}_{ij}}\to \] represent the position of element a in the given matrix

e.g \[A=[{{a}_{ij}}]\] & if         

 

• Order of Matrix: It is the symbol which represent that how many no. of rows and no. of columns the matrix has

In the above example

Order of matrix \[A=3\times 2\] in which 3 determine no. of row & 2 determine no. of column of given matrix.

 

• Operation of Matrix:

(a) Addition of matrices              

(b) Subtraction of matrix

(c) Multiplication of matrix                      

(d) Adjoint of matrix

(e) Inverse of matrixes.

 

• Addition of Matrices: Let \[A={{[{{a}_{ij}}]}_{\,m\,\times \,n}}\] & \[B={{[{{b}_{ij}}]}_{\,m\,\times \,n}}\]be two matrices, having same order. Then A + B or B + A is a matrix whose elements be formed through corresponding addition of elements of two given matrices

                                                         

• Subtraction of Matrices: The subtraction of a matrix takes place in the similar manner as the addition. But \[A-B\ne B-A\] i.e. \[A-B=-\,(B-A)\]

Note: For addition or substraction operation of two or more than two matrices, the given matrices should be of the same order.

 

• Multiplication operation: Let \[A={{[{{a}_{ij}}]}_{\,m\,\times \,k}}\] a matrix of order \[m\times k\]& \[B={{[{{b}_{ij}}]}_{\,k\,\times \,p}}\]is a matrix of order\[k\times p\]. For multiplication of two matrices, no. of columns of 1st matrix should be equal to no. of rows of 2nd matrix. Otherwise these matrices cannot be multiplied.

\[A\times B=[{{c}_{ij}}]\] be a matrix whose order will be \[mxp.\]

e.g. \[A{{\left[ \xrightarrow[5\,\,\,\,\,\,2\,\,\,\,\,\,1]{2\,\,\,\,\,\,1\,\,\,\,\,\,3} \right]}_{\,2\,\times \,3}}\] &

                                                                           

             

Hence \[A\times B\ne B\times A.\]

 

• Types of Matrices:

(a) Row Matrix: A matrix having order \[1\times n\] i.e. which has only one row, is said to be row matrix.

e.g. \[A={{[1\,\,2\,\,3\,\,4\,\,5]}_{\,1\,\times \,5}}\]

(b) Column Matrix: A matrix, having order \[m\times 1\] i.e. which has only one column is said to be column matrix.

(c) Square Matrix: A matrix having same no. of rows & columns is said to be square matrix.

(d) Diagonal Matrix: A square matrix is said to be diagonal matrix if all of its non-diagonalelements are zero.

 i.e  \[{{a}_{ij}}=0\,\,if\,\,i\ne \,j\,=\,\,something\,\,if\,i=j\]

• Transpose of a matrix: Let A is a matrix of order mxn. Then the transpose of a matrix A is a matrix of order nxm which is obtained by interchanging the rows& columns of matrix A. It is denoted by \[A'\]or \[{{A}^{c}}\]

 

 

 

• Properties of Transpose of a matrix:

(i) If A and B be the two matrix of the same order then \[{{(A\,\pm B)}^{c}}={{A}^{c}}\pm {{B}^{c}}\,\,\,\,\,\,\,\,\,\,\,m\times n\]

(ii) If A be the matrix and k be any scalar quantity then \[{{(kA)}^{c}}=k.{{A}^{c}}\]

(iii) If A & B be two matrix of order \[m\times n\] and \[n\times p\] respectively, there exists the multiplication between A & B. Then

\[{{(A\times B)}^{c}}={{B}^{c}}.\,{{A}^{c}}\]

(iv) The double transpose of any matrix gives its primal matrix i.e. \[{{({{A}^{c}})}^{c}}=A.\]

 

• Symmetric Matrix: A square matrix A is said to be symmetric if the matrix A Is equal to its transpose matrix.

i.e. \[A=A'\]

The matrix,\[A={{[{{a}_{ij}}]}_{\,m\,\times \,n}}\] is said to symmetric matrix if \[{{a}_{ij}}={{a}_{ji}}.\]

 

 

 

• Skew Symmetric Matrix: A square matrix A is said to be the skew symmetric if the matrix A is equal to its transpose matrix with negative sign.

i.e. \[A=-A'\]

The matrix, \[A={{[{{a}_{ij}}]}_{\,m\,\times \,m}}\] is said to symmetric matrix if

e.g. \[{{a}_{ij}}=-{{a}_{ji}}\,\,\,\forall \,\,i\And j\]

 

 

Note: Every square matrix is written as the sum of the symmetric & skew symmetric matrices.

i.e. If A is a square matrix then \[A=\frac{1}{2}[(A+{{A}^{c}})+(A-{{A}^{c}})]\]

Where \[A+{{A}^{c}}=\] symmetric matrix & \[A-{{A}^{c}}=\] skew symmetric matrix.

 

• Properties of symmetric & skew symmetric matrices.

(i) If A &B be two symmetric (or skew symmetric) matrices, then A + B will be symmetric matrix (or skew symmetric matrix).

(ii) If A is symmetric (or skew symmetric) matrix and k is any scalar quantity, then (kA) is a symmetric (or skew symmetric) matrix.

(iii) If A and B are symmetric matrices of the same order, the product AB is symmetric iff AB = BA

(iv) The matrix B' AB is symmetric or skew symmetric accordingly as A is symmetric or skew symmetric.

(v) If A and B are symmetric matrices of the same order, then

(a) \[AB-BA\]is a skew symmetric matrix.

(b) \[AB+BA\]is a symmetric matrix.

(vi) All positive integral powers of a symmetric (or skew symmetric) matrix will be symmetric (or skew-symmetric) matrix.

 

• Singular Matrix: A square matrix is said to be singular matrix if the determinant of A i.e. | A | is zero. i.e. | A | = 0. Otherwise, it is a non-singular matrix.

 

• Adjoint of a square matrix: Let \[A=[{{a}_{ij}}]\] be a square matrix of order m and B be a square matrix order m whose element be the corresponding co-factors of the elements of matrix A, then the transpose of B is said to be the adjoint of the matrix A.

In \[{{R}_{1}}\]row,

co-factor of 1 i.e. \[{{a}_{11}}=3\]

co-factor of \[-1\]i.e. \[{{a}_{12}}=-12\]

co-factor of 2 i.e. \[{{a}_{13}}=6\]

In \[{{R}_{2}}\]row,

co-factor of 2 i.e. \[{{a}_{21}}=-(-1)=1\]

co-factor of 3 i.e. \[{{a}_{22}}=5\]

co-factor of 5 i.e. \[{{a}_{23}}=-(-2)=2\]

In \[{{R}_{3}}\]row,

co-factor of \[-2\]i.e. \[{{a}_{31}}=-11\]

co-factor of 0 i.e. \[{{a}_{32}}=-(5-4)=-1\]

co-factor of 1 i.e. \[{{a}_{33}}=3+2=5\]

 

 

Properties of adjoint of a matrix:

(i) If A is the square matrix of order m, then adjoint A' = (adj of A)

(ii) If A & B be two the square matrices of the same order, then adj(AB) = (adj A).(adj B)

(iii) The adjoint of a diagonal matrix is a diagonal matrix.

(iv) adj(adj A) \[=\,\,|A|{{\,}^{n\,-\,2}}.\] A where A is the non-singular matrix

 

• Inverse of the square of a matrix: Let A is the square matrix of order\[m\times m\], then the square matrix B of order \[m\times m\] is said to be the inverse of the matrix A such that

AB = BA = I

Where I is the unit matrix.

i.e. It is represent by \[{{A}^{-1}}\]

So, \[{{A}^{-1}}=\frac{\alpha dj\,\,of\,\,A}{|A|}\]

i.e. A.\[{{A}^{-1}}={{A}^{-1}}.\] \[A=I\]

 

• Properties of inverse matrix:

(i) The inverse of the square of a matrix exists iff A is non-singular.

(ii) Double inverse of the square matrix is primal matrix A itself, i.e. \[{{({{A}^{-1}})}^{-1}}=A\]

(iii) If A & B be two invertible matrices of the same order m x m then AB is also invertible & moreover\[{{(AB)}^{-1}}={{B}^{-1}}{{A}^{-1}}\]

(iv) The inverse of the transpose of a matrix A is equal to the transpose of the inverse of matrix A.

i.e. \[{{(A')}^{-1}}=({{A}^{-1}})'\]

(v) If A is the symmetric matrix such that\[|A|\,\,\ne 0\], then \[{{A}^{-1}}\] is also the symmetric matrix.

Solved Examples

1. If A B = A and B A = B than find the value of \[{{B}^{2}}\]

Sol.  \[\therefore \,\,\,BA=B\]

Right multiply B on both such

\[\therefore \,\,(B\,A)B=B.\,\,\,B\] \[B.(A\,B)={{B}^{2}}\]

\[\Rightarrow B.\,\,A={{B}^{2}}\,\,\,\,\,\,\,\,\,[\because A\,\,B=A]\]

\[\Rightarrow B={{B}^{2}}\]

\[\Rightarrow {{B}^{2}}=B\]

 

2. If\[A={{({{a}_{ij}})}_{\,2\,\times \,2}}\], where \[{{a}_{ij}}=i+j\], find A.

            Sol.

 

 

3. Find the value of

            Sol.   

 

 

                        In 2nd put \[{{R}_{1}}\to a.{{R}_{1}}\] \[{{R}_{2}}\to b{{R}_{2,}}{{R}_{3}}\to c{{R}_{3}}\]