Introduction : Matrix is an arrangement of objects in rows and column.
Vector
A vector is a matrix of order \(3\) by \(1\).
Example : \[V=\left(\begin{array}{ccc}1 \\2 \\3 \\\end{array}\right)\]
Matrix :
A rectangular array of \(m*n\) elements into \(m-\)rows and \(n\)-columns is called a matrix of order \(m\) by \(n\).
An \( (i, j) \) entry of a matrix which is at position \(i_{th}\)-row and \(j_{th}\)-column is denoted by \( a(i, j) \).
Example (1) : A matrix of order \(2\) by \(3\) is
\[A=\left(\begin{array}{ccc}3 & -2 & 1 \\4 & -5 & 8\\\end{array}\right)\]
it’s entries are \( a(1, 1) = 3, a(1, 2) = -2, a(1, 3) = 1, a(2, 1)= 4, a(2, 2) =-5, a(2, 3) = 8\)
Example (2) : A matrix of order \(3\) by \(3\) is
\[B=\left(\begin{array}{ccc}2 & 3 & 4 \\3 & 2 & 1 \\4 & 6 & 8 \\\end{array}\right)\]
Algebra of Matrices
Addition of Matrices :
Addition of two matrices are possible if and only if these matrices are of the same order and resultant matrix will be of same order.
Example : Let \[A=\left(\begin{array}{ccc}0 & -1 & 2 \\-3 & 2 & 0 \\4 & 2 & 0 \\\end{array}\right), B=\left(\begin{array}{ccc}9 & 6 & 3 \\-5 & 0 & 2 \\3 & 3 & 2 \\\end{array}\right)\]
then addition of A and B is \[A+B=\left(\begin{array}{ccc}0+9 & -1+6 & 2+3 \\-3-5 & 2+0& 0+2\\ 4+3&2+3&0+2\\\end{array}\right)=\left(\begin{array}{ccc}9 & 5 & 5 \\-8 & 2& 2\\ 7&5&2\\\end{array}\right) \]
Subtraction of Matrices
Subtraction of a matrix from other is defined if and only if these matrices are of the same order and resultant matrix will be of same order also.
\[A-B=\left(\begin{array}{ccc}0-9 & -1-6 & 2-3 \\-3+5 & 2-0& 0-2\\ 4-3&2-3&0-2\\\end{array}\right)=\left(\begin{array}{ccc}-9 & -7 & -1 \\2 & 2& -2\\ 1&-1&-2\\\end{array}\right) \]
Scalar multiplication :
When a scalar \(k\) is multiplied with each entry \( a(i, j) \) of matrix \(A\) then it is called scalar multiplication of k with A.
Example : Suppose \(k = 5\) and \(A=\left(\begin{array}{ccc}2 & -1 & 1 \\0 & 4 & 2 \\0 & -3 & -5 \\\end{array}\right)\) then scalar multiplication of k with A is
\[kA=5A = \left(\begin{array}{ccc}2*5 & -1*5 & 1*5 \\0*5 & 4*5 & 2*5 \\0*5 & -3*5 & -5*5 \\\end{array}\right) = \left(\begin{array}{ccc}10 & -5 & 5 \\0 & 20 & 10\\0 & -15 & -25 \\\end{array}\right)\]
Product of Matrices :
Product of two matrices \(A, B\) is possible if number of columns in \(A\) is equal to number of rows in \(B\).
Example (1)
If \(A =\left(\begin{array}{cc}2 & 8 \\3 & 4 \end{array}\right), B =\left(\begin{array}{cc}-3 & 1 \\0 & -1 \end{array}\right)\)
then find the product \(AB\)
Solution:
\[AB =\left(\begin{array}{cc}2 & 8 \\3 & 4 \end{array}\right)\left(\begin{array}{cc}-3 & 1 \\0 & -1 \end{array}\right)\]
\[=\left(\begin{array}{cc}2*(-3)+8*0 & 2*1+8*(-1) \\3*(-3)+4*0 & 3*1+4*(-1) \end{array}\right) \]
\[AB=\left(\begin{array}{cc}-6 & -6 \\-9 & -1 \end{array}\right)\]
Example (2)
Multiply matrix \(A =\left(\begin{array}{ccc}4 & -1 & 2 \\3 & 4 & 3 \\\end{array}\right)\) with matrix \(B =\left(\begin{array}{cc}-5 & 1 \\-2 & -1 \\1 & 1 \\\end{array}\right)\)
Solution:
\[AB =\left(\begin{array}{ccc}4 & -1 & 2 \\3 & 4 & 3 \\\end{array}\right)\left(\begin{array}{cc}-5 & 1 \\-2 & -1 \\1 & 1 \\\end{array}\right)\]
\[\left(\begin{array}{ccc}4 (-5) +(-1 )(-2)+ 2*1 &4*1+(-1)( -1) + 2(1)\\3 (-5) +4(-2)+ 3*1 &3*1+4( -1) + 3(1) \\\end{array}\right)\]
\[ = \left(\begin{array}{cc}-16 & 7 \\-20 & 2 \\\end{array}\right)\]
Transpose of a Matrix
Matrix obtained by interchanging rows and columns of a matrix \(A\) is called transpose of \(A\). It is denoted by \(tr(A)\) or \(A^{T}\).
Example (1)
If \( A =\left(\begin{array}{cc}9 & 1 \\0 & 5 \end{array}\right)\) then \(A^{T} =\left(\begin{array}{cc}9 & 0 \\1 & 5 \end{array}\right)\)
Example (2)
Let \( A =\left(\begin{array}{ccc}1 & 2 & 1 \\0 & -1 & 2 \\3 & 1 & 0 \end{array}\right)\) then \(A^T =\left(\begin{array}{ccc}1 & 0 & 3 \\2 & -1 & 1 \\1 & 2 & 0 \end{array}\right)\)
Singular Matrix
A square matrix whose determinant is zero is called singular. A singular matrix always has linearly dependent rows or columns.
Examples
\(\left(\begin{array}{cc}5 & -1 \\15 & -3 \end{array}\right)\). In this matrix second row is two times row first.</li><li>\(\left(\begin{array}{ccc}1 & -1 & 1 \\3 & 0 & 4 \\2 & -2 & 2 \end{array}\right)\). In this matrix third row is three times of row first.
are singular matrices.
Non-Singular Matrix
A square matrix with non-zero determinant is called non-singular. In a non-singular matrix all rows or columns must be linearly independent.
Examples
\(\left(\begin{array}{cc}0 & -1 \\4 & -3 \end{array}\right)\)
\(\left(\begin{array}{ccc}6 & 7 & -1 \\4 & -4 & 1 \\0 & 3 & 2 \end{array}\right)\).
are non- singular matrices.
Invertible Matrix
A non-singular matrix is called invertible matrix.
Examples
\(\left(\begin{array}{cc}2 & -1 \\3 & 1 \end{array}\right)\) is invertible since determinant \(|A| =5 \ne 0\)
\(\left(\begin{array}{ccc}5 & 4 & -1 \\3 & -4 & 1 \\6 & 3 & 0 \end{array}\right)\) is invertible since determinant \(|A| =-24\ne 0\)
Submatrix of a Matrix
Let \(A = \left(a(i,j)\right)_{1\le i, j\le 1}\) be a square matrix of order \(n\) then it’s submatrix \(S_{i,j}\) of order \(n-1\) is a matrix obtained by deleting \(i_{th}\) row and \(j_{th}\) column of \(A\).
Example (1) Let \(A =\left( {\begin{array}{*{20}{c}}1&3\\2&0\end{array}} \right)\) then there are four submatrices given as follows
\(S_{1,1}=(0),S_{1,2}=(2),S_{2,1}=(3),S_{2,2} =(1)\).
Example (2) If \(A=\left( {\begin{array}{*{20}{c}}2&3&-1\\4&0&5\\6&0&-2\end{array}} \right)\) then it’s submatrices are
\(S_{1,1} = \left( {\begin{array}{*{20}{c}}0&5\\0&-2\end{array}} \right),S_{1,2} =\left( {\begin{array}{*{20}{c}}4&5\\6&-2\end{array}} \right),S_{1,3} = \left( {\begin{array}{*{20}{c}}4&0\\6&0\end{array}} \right),S_{2,1}=\left( {\begin{array}{*{20}{c}}3&-1\\0&-2\end{array}} \right)\)
\(S_{2,2}=\left( {\begin{array}{*{20}{c}}2&-1\\6&-2\end{array}} \right),S_{2,3}=\left( {\begin{array}{*{20}{c}}2&3\\6&0\end{array}} \right),S_{3,1}=\left( {\begin{array}{*{20}{c}}3&-1\\0&5\end{array}} \right),S_{3,2}=\left( {\begin{array}{*{20}{c}}2&-1\\4&5\end{array}} \right),S_{3,3}=\left( {\begin{array}{*{20}{c}}2&3\\4&0\end{array}} \right)\)
Minor of Matrix
.\((i,j)\) minor of a matrix \(A\) is denoted as \(M_{i,j}\) and it is the determinant of submatrix \(S_{i,j}\) of \(A\).
Example For \(A=\left( {\begin{array}{*{20}{c}}2&3&-1\\4&0&5\\6&0&-2\end{array}} \right)\) it’s minors are
\(M_{1,1} = det\left( S_{1,1}\right) =0,M_{1,2} =det\left( S_{1,2}\right)=-38,M_{1,3} = det\left( S_{1,3}\right)=0,M_{2,1}=det\left( S_{2,1}\right)=-6\)
\(M_{2,2}=det\left( S_{2,2}\right)=2,M_{2,3}=det\left( S_{2,3}\right)=-18,M_{3,1}=det\left( S_{3,1}\right)=15,M_{3,2}=det\left( S_{3,2}\right)=14,M_{3,3}=det\left( S_{3,3}\right)=-12\)
Cofactor of Matrix
Cofactor of a matrix \(A\) is a matrix \(c_{i,j}\) where \(c_{i,j} =(-1)^{i+j}M_{i,j}\). It is denoted as \(cof(A)\).
Example \[cof \left( {\begin{array}{*{20}{c}}2&3&-1\\4&0&5\\6&0&-2\end{array}} \right)=\left( {\begin{array}{*{20}{c}}0&38&0\\6&2&18\\15&-14&-12\end{array}} \right)\]
Adjoint of Matrix
Transpose of cofactor of a matrix \(A\) is called adjoint of \(A\). It is denoted as \(adj(A)\).
\[adj(A) = tr\left( {\begin{array}{*{20}{c}}0&38&0\\6&2&18\\15&-14&-12\end{array}} \right) =\left( {\begin{array}{*{20}{c}}0&6&15\\38&2&-14\\0&18&-12\end{array}} \right)\]
Inverse of Matrix
Ratio of \(adj(A)\) to \(det(A)\) is called inverse of \(A\). It is denoted as \(A^{-1}\) or \(inv(A)\).
\[A^{-1}=\frac{adj(A)}{det(A)}\]
Example
For \(A=\left( {\begin{array}{*{20}{c}}2&3&-1\\4&0&5\\6&0&-2\end{array}} \right)\)
Since \(det(A) =114\)
hence \(A^{-1} =\frac{1}{114}\left( {\begin{array}{*{20}{c}}0&6&15\\38&2&-14\\0&18&-12\end{array}} \right)\)
Types of Matrices
Zero Matrix
Zero matrix is matrix will all entries zero. It is denoted as \( O\).
Example A zero matrix of order two is
\[ O =\left( {\begin{array}{*{20}{c}}0&0\\0&0\end{array}} \right)\]
Identity Matrix
An identity matrix \(I\) is a matrix whose all entries on it’s main diagonal are one and all other entries are zeros.
Example Identity matrix of order three is
\[I =\left( {\begin{array}{*{20}{c}}1&0&0\\0&1&0\\0&0&1\end{array}} \right)\]
Square Matrix
A matrix that has equal number of rows and columns.
Examples If \(A =\left( {\begin{array}{*{20}{c}}1&0\\2&1\end{array}} \right), B =\left( {\begin{array}{*{20}{c}}1&0&3\\2&1&1\\1&1&1\end{array}} \right)\) are square matrices of order two and three respectively.
Diagonal Matrix
A matrix whose all entries are zero except on it’s main diagonal is called diagonal matrix.
Example \(D =\left( {\begin{array}{*{20}{c}}1&0&0\\0&2&0\\0&0&3\end{array}} \right)\)
Triangular Matrix
A square matrix \(A=(a_{i,j})_{1 \le i,j \le n}\) whose entries satisfy either \(a_{i,j} =0 \forall i<j\) or \(a_{i,j} =0 \forall i>j\) is called a triangular matrix. Therefore triangular matrices are divided into following two types
Upper Triangular Matrix
If A square matrix \(A=(a_{i,j})_{1 \le i,j \le n} \) satisfies \(a_{i,j} =0 \forall i<j\) is called upper triangular.
\[A =\left( {\begin{array}{*{20}{c}}1&{ – 1}&0\\0&9&{ – 2}\\0&0&0\end{array}} \right)\]
Example
Lowe Triangular Matrix
If A square matrix \(A=(a_{i,j})_{1 \le i,j \le n} \) satisfies \(a_{i,j} =0 \forall i>j\) is called lower triangular.
Example
\[A=\left( {\begin{array}{*{20}{c}}1&0&0\\3&{ – 2}&0\\0&2&4\end{array}} \right)\]
Symmetric Matrix
A matrix \(A\) whose transpose \(tr(A)\) is same as as \(A\) is called symmetric.
\[A=A^{T}\]
Example (1)
If \(A =\left(\begin{array}{ccc}1 & 0 & 1 \\0 & 1 & 1 \\1 & 1 & 0 \end{array}\right)\) then \(A^T =\left(\begin{array}{ccc}1 &0 & 1 \\0 & 1 & 1 \\1 & 1 & 0 \end{array}\right)\)
Clearly \(A = A^T\).
Skew-Symmetric Matrix
A matrix \(A\) whose transpose \(tr(A)\) is negative times \(A\) is called skew-symmetric.
\[A=-A^{T}\]
Example
Hermitian Matrix
Skew-Hermitian Matrix
Definite Matrix
Positive Definite Matrix
Negative Definite Matrix
Orthogonal Matrix
Unitary Matrix
Normal Matrix
Idemptotant Matrix
Nilpotent Matrix