A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements or the entries of the matrix, and the general form of a matrix is given by:
In general, we represent the element in row i and column j as aij .
A matrix having m rows and n columns is called a matrix of order m × n or simply m × n matrix (read as an m by n matrix). Further, the number of elements in an m × n matrix will be equal to mn.
Types of Matrices
1. Row Matrix
A matrix is said to be a row matrix if it has only one row. For example:
In general, a row matrix is of the order 1 × n.
2. Column Matrix
A matrix is said to be a column matrix if it has only one column. For example:
In general, a column matrix is of the order m × 1.
3. Square Matrix
A matrix in which the number of rows are equal to the number of columns, is said to be a square matrix. Thus an m × n matrix is said to be a square matrix if m = n and is known as a square matrix of order ‘n’. For example:
The elements (entries) a11, a22, …, ann are said to constitute the diagonal of a matrix. Here, the diagonal elements of A are: 1, 0, 1.
4. Diagonal Matrix
A square matrix is said to be a diagonal matrix if all its non diagonal elements are zero. For example:
5. Scalar Matrix
A diagonal matrix is said to be a scalar matrix if its diagonal elements are equal. For example:
6. Identity Matrix
A square matrix in which elements in the diagonal are all 1 and rest are all zero is called an identity matrix. We denote the identity matrix of order n by In. Also, when the order is clear from the context, we simply write it as I. For example:
7. Zero Matrix
A matrix is said to be zero matrix or null matrix if all its elements are zero. We denote zero matrix by O. For example:
Equality of Matrices
Two matrices A = [aij] and B = [bij] are equal if:
1.They are of the same order, and
2.Each element of A is equal to the corresponding element of B, that is aij = bij for all i and j.
Operations on Matrices
Addition of Matrices
In general, if A = [aij] and B = [bij] are two matrices of the same order, say m × n. Then, the sum of the two matrices A and B is defined as a matrix C = [cij]m × n, where cij = aij + bij, for all possible values of i and j. For example:
If A and B are not of the same order, then we cannot define A + B.
Properties of Matrix Addition:
1.Commutative Law: A+B=B+A.
2.Associative Law: (A+B)+C=A+(B+C).
3.Additive Identity: A+O=O+A=A.
4.Additive Inverse: A+(-A)=(-A)+A=O.
Multiplication of a matrix by a scalar
If A = [aij] m × n is a matrix and k is a scalar, then kA is another matrix which we obtain by multiplying each element of A by the scalar k. For example:
Properties of scalar multiplication of a matrix
1.k(A+B) = kA + kB.
2.(k+p)A = kA+pA.
Multiplication of matrices
The product of two matrices A and B exists if the number of columns of A is equal to the number of rows of B.
Let A = [aij] be an m × n matrix and B = [bjk] be an n × p matrix. Then the product of the matrices A and B is the matrix C of order m × p. To get the (i, k)th element cik of the matrix C, we take the ith row of A and kth column of B, multiply them elementwise and take the sum of all these products.
If AB is defined, BA need not be defined.
If both A and B are square matrices of the same order, then both AB and BA are defined.
Even if AB and BA are both defined, it is not necessary that AB = BA.
We can also obtain a zero matrix by multiplying two non-zero matrices.
Properties of multiplication of matrices
1.Associative Law: (AB)C = A(BC).
(i) A(B+C) = AB + AC.
(ii) (A+B)C = AC + BC.
3.Multiplicative Identity: IA = AI = A.
Transpose of a Matrix
If A = [aij] be an m × n matrix, then the matrix obtained by interchanging the rows and columns of A is called the transpose of A. We denote transpose of the matrix A by A’. For example:
Properties of transpose of the matrices
1.(A’)’ = A.
2.(kA)’ = kA’
3.(A+B)’ = A’ + B’
4.(AB)’ = B’A’
Symmetric and Skew Symmetric Matrices
Symmetric Matrix: A is symmetric if A’ = A.
Skew Symmetric Matrix: A is skew symmetric if A’ = -A.
For any square matrix A with real number entries, A + A’ is a symmetric matrix and A – A’ is a skew symmetric matrix.
We can express any square matrix as the sum of a symmetric and a skew symmetric matrix.
Elementary Operation (Transformation) of a Matrix
There are six operations (transformations) on a matrix, three of which are due to rows and three due to columns, which are known as elementary operations or transformations.
(i) The interchange of any two rows or two columns. Symbolically the interchange of ith and jth rows is denoted by Ri→Rj and similarly interchange of ith and jth column is denoted by Ci→Cj.
(ii) The multiplication of the elements of any row or column by a non zero number. Symbolically, the multiplication of each element of the ith row by k, where k ≠ 0 is denoted by Ri→ k Ri.
(iii) The addition to the elements of any row or column, the corresponding elements of any other row or column multiplied by any non zero number. Symbolically, the addition to the elements of ith row, the corresponding elements of jth row multiplied by k is denoted by Ri →Ri + kRj.
If A is a square matrix of order m, and if there exists another square matrix B of the same order m, such that AB = BA = I, then we define B as the inverse matrix of A and we denote it by A– 1. In that case, we say that A is invertible.
The inverse of a square matrix, if it exists, is unique.
If A and B are invertible matrices of the same order, then
(AB)– 1 = B– 1A– 1.
Inverse of a matrix by elementary operations
Apply row transformation on A=IA to get I=BA, so B is the inverse of A.
Similarly, apply column transformation on A=AI to get I=AB, so B is the inverse of A.
Further, if during these transformations, we obtain all zeros in one or more rows/columns of the matrix on LHS, then A– 1 does not exist.
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