Chapter 3
MATRICES
Top Terms
- A matrix is an ordered rectangular array of numbers (real or complex) or functions or names or any type of data. The numbers or functions are called the elements or the entries of the matrix.
- The horizontal lines of elements constitute the rows of the matrix and the vertical lines of elements constitute the columns of the matrix.
- Each number or entity in a matrix is called its element.
- If a matrix contains m rows and n columns, then it is said to be a matrix of the order m × n (read as m by n).
- The total number of elements in a matrix is equal to the product of its number of rows and number of columns.
- A matrix is said to be a column matrix if it has only one column.
- A = [aij]m × 1 matrix is said to be a row matrix if it has only one row.
- A matrix is said to be a row matrix if it has only one row.
- B = [bij]1 × n is row matrix of order 1 × n.
- Rectangular matrix: A matrix in which the number of rows is not equal to the number of columns is called a rectangular matrix.
- A matrix each of whose elements is zero is called a zero matrix or null matrix.
- A matrix in which the number of rows is equal to the number of columns is said to be a square matrix. A matrix of order ‘m × n’ is said to be a square matrix if m = n and is known as a square matrix of order ‘n’.
- A square matrix which has every non—diagonal element as zero is called a diagonal matrix.
- A square matrix A = [aij]m × m is said to be a diagonal matrix if all its non-diagonal elements are zero, i.e., a matrix A = [aij]m × m is said to be a diagonal matrix if aij = 0 when i ≠ j.
- A square matrix in which the elements in the diagonal are all 1 and the rest are all zero is called an identity matrix. A square matrix A = [aij]n × n is an matrix if
16. A diagonal matrix is said to be a scalar matrix if its diagonal elements are equal, that is a square matrix B = [bij]n × n is said to be a scaler matrix if bij = 0 when i ≠ j and bij = k when i = j for some constant k.
17. Upper triangular matrix: A square matrix A = [aij] is called an upper triangular matrix if aij = 0 for all i > j. In an upper triangular matrix, all elements below the main diagonal are zero.
18. Lower triangular matrix: A square matrix A = [a„] is called a lower triangular matrix if aij = 0 for all i < j. In a lower triangular matrix, all elements above the main diagonal are zero.
19. Two matrices are said to be equal if they are of the same order and have the same corresponding elements.
20.Two matrices A = [aij] and B = [bij] are said to be equal if they are of the same order. Each element of A is equal to the corresponding element of B, that is aij = bij for all i and j.
21. If A is a matrix, then its transpose is obtained by interchanging its rows and columns. Transpose of a matrix A is denoted by At. If A = [aij] be an n × m matrix, then the matrix obtained by interchanging the rows and columns of A is called the transpose of A. Transpose of the matrix A is denoted by A’ or (AT). Theft is, (AT)ij = aji for all l = 1, 2, … m; j = 1, 2, … n.
22. If A = [aij]nxn is an n × n matrix such that AT = A, then A is called a symmetric matrix. In a symmetric matrix, aij = aji for all i and j.
23. If A = [aij]nxn is an n × n matrix such that AT = -A, then A is called a skew-symmetric matrix. In a skew-symmetric matrix, aij = aji.
24. All main diagonal elements of a skew-symmetric matrix are zero.
25. Every square matrix can be expressed as the sum of a symmetric and a skew-symmetric matrix.
26. All positive integral powers of a symmetric matrix are symmetric.
27. All odd positive integral powers of a skew-symmetric matrix are skew-symmetric.
28. Let A and B be two square matrices of the order n such that AB = BA = I.
Then A is called the inverse of B and is denoted by B = A-1. If B is the inverse of A, then A is also the inverse of B.
29. If A and B are two invertible matrices of the same order, then (AB)-1 = B-1 A-1.
Top Concepts
- The order of a matrix gives the number of rows and columns present in the matrix.
- If a matrix A has m rows and n columns, then it is denoted by A = [aij]m × n. Here aij is i-j th or (i, j)th element of the matrix.
- The simplest classification of matrices is based on the order of the matrix.
- In case of a square matrix, the collection of elements a11. a22 and so on constitute the Principal Diagonal or simply the diagonal of the matrix.
- The diagonal is defined only in the case of square matrices.
6. Two matrices of the same order are comparable matrices.
7. If A = [aij]m × n and B = = [bij]m × n are two matrices of the order m × n, then their sum is defined as a matrix C = [cij]m × n where cij = aij + bij for 1≤i≤m,1≤j≤n.
8. Two matrices can be added (or subtracted) if they are of the same order.
9. For multiplying two matrices A and B, the number of columns in A must be equal to the number of rows in B.
10. If A = [aij]m × n is a matrix and k is a scalar, then kA is another matrix which is obtained by multiplying each element of A by the scalar k.
Hence, kA = [kaij] m × n
11. If A = [aij]m × n and B = = [bij]m × n are are two matrices, then their difference is represented as A – B = A + (-1)B.
12. Properties of matrix addition
- Matrix addition is commutative, i.e., A + B = B + A
- Matrix addition is associative, i.e. (A + B) + C = A + (B + C)
- Existence of additive identity: NuII matrix is the identity with respect to addition of matrices.
Given a matrix A = [aij]m × n, there will be a corresponding null matrix O of the same order such that A + O = O + A = A
- The existence of additive inverse: Let A = [aij]m × n be any matrix, then there exists another matrix -A = -[aij]m × n Such that
A + (-A) = (-A) + A = O.
13. Cancellation law: If A, B and C are three matrices of the same order, then
14. Properties of scalar multiplication of matrices
If A = [aij], B = [bij] are two matrices, and k and L are real number, then
- k(A + B) = kA + kB
- (k + l)A = kA + IA
- k(A + B) = k([aij]+[bij]) = k[aij] + k[bij] = kA + kB
- (k + L)A = (k + L) [aij] = [(k + L)aij] = k[aij] + L[aij] = kA + LA
15. If A = [aij]mxp, B = [bij]pxn are two matrices, then their product AB is given by C = [cij]mxn such that
In order to multiply two matrices A and B, the number of columns in A = number of rows in B.
16. Properties of Matrix Multiplication
Commutative law does not hold in matrices, whereas associative and distributive laws hold for matrix multiplication.
- In general, AB ≠ BA
- Matrix multiplication is associative A(BC) = (AB)C
- Distributive laws:
A(B + C) = AB + AC
(A + B)C = AC + BC
17. The multiplication of two non-zero matrices can result in a null matrix.
18. If A is a square matrix, then we define A1 = A and An+1 = An. A
19. If A is a square matrix, a0, a1, a2, …, an are constants, then a0An + a1An-1 + a2An-2 + … + an-1A + an is called a matrix polynomial.
20. If A, B and C are matrices, then AB = AC, A ≠ 0 ⇒ B = C.
In general, the cancellation law is not applicable in matrix multiplication.
21. Properties of transpose of matrices
- If A is a matrix, then (AT )T = A
- (A + B)T = AT + BT
- (kB)T = kBT, where k is any constant.
22. If A and B are two matrices such that AB exists, then (AB)T = BT AT.
23. If A, B and C are two matrices such that AB exists, then (ABC)T = CT BT AT.
24. Every square matrix can be expressed as the sum of a symmetric and a skew-symmetric
25. A square matrix A is called an orthogonal matrix when AAT = ATA = I.
26. A null matrix is both symmetric and skew symmetric.
27. Multiplication of diagonal matrices of the same order will be commutative.
28. There are six elementary operations on matrices—three on rows and three on columns. The first operation is interchanging the two rows, i.e., Ri Rj implies that the ith row is interchanged with the jth row. The two rows are interchanged with one another and the rest of the matrix remains the same.
29. The second operation on matrices is to multiply a row with a scalar or a real number, i.e., Ri kRi that ith row of a matrix A is multiplied by k.
30. The third operation is the addition to the elements of any row, the corresponding elements of any other row multiplied by any non-zero number, i.e., Ri → Ri + kRi k multiples of the jth row elements are added to the ith row elements.
31. Column operation on matrices ar,
- Interchanging the two columns: Cr ↔ Ck indicates that the rth column is interchanged with the kth column.
- Multiply a column with a non-zero constant, i.e., Ci → kCi
- Addition of a scalar multiple of any column to another column, i.e. Ci → Cj + kCj
32. Elementary operations help in transforming a square matrix to an identity matrix.
33. The inverse of a square matrix, if it exists, is unique.
34. The inverse of a matrix can be obtained by applying elementary row operations on the matrix A = IA. In order to use column operations, write A = AI.
35. Either of the two operations—row or column—can be applied. Both cannot be applied simultaneously.
36. For any square matrix A with real number entries, A + A’ is a symmetric matrix and A — A’ is a skew- symmetric matrix.
Laws of algebra are not applicable to matrices, i.e.
(A + B)2 ≠ A2 + 2AB + B2
and
(A + B) (A – B) ≠ A2 – B2
Top Formulae
- An m × n matrix is a square matrix if m = n.
- A = [aij] = [bij] = B if
- A and B are of the same order, (ii) aij = bij for all possible values of i and j.
3. kA = k[aij]m × n = [k(aij)]m × n
4. –A = (–1) A
5. A – B = A + (–1)B
6. If A = [aij]m × n and B = [bij]n × p, then AB = C = [cik]m × p,
7. Elementary operations of a matrix are as follows:
- Ri Rj or Ci ↔ Cj
- Ri → kRi or Ci → kCi
- Ri → Ri + kRj or Ci → Ci + kCj
