CBSE Class 12 Mathematics Matrices And Determinants Notes Set 01

ALGEBRA OF MATRICES

BASIC CONCEPTS

• Matrix: A matrix is a rectangular arrangement of numbers or functions arranged into a fixed number of rows and columns.

  A matrix is written inside brackets [ ]. Each entry in a matrix is called an element of the matrix.

  Order of Matrix: The dimension or order of matrix is defined by the number of rows and columns of that matrix. By conversion the dimension or order of a matrix is given by

  No. of rows \( \times \) No. of columns

  If a matrix have \( m \) rows and \( n \) columns then its order (dimension) is written as \( m \times n \) and read as \( m \) by \( n \).

• Row Matrix: A matrix having one row and any number of column is called a row matrix. In other words, matrix of order \( 1 \times n \) is always a row matrix.

  e.g., \( [a, b, c, d]_{1 \times 4} \) is a row matrix.

• Column Matrix: A matrix having any number of rows but only one column is called column matrix. In other words, a matrix of order \( m \times 1 \) is always a column matrix.

  e.g., \( \begin{bmatrix} a \\ b \\ c \\ d \end{bmatrix}_{4 \times 1} \) is a column matrix.

• Square Matrix: A matrix in which the number of rows is equal to the number of columns, say \( n \), is called a square matrix of order \( n \).
• Diagonal Elements: The elements \( a_{ij} \) of a square matrix \( A = [a_{ij}]_{n \times n} \) for which \( i = j \), i.e., the elements \( a_{11}, a_{22}, ..., a_{nn} \) are called the diagonal elements and the line along which the diagonal elements lie, is called the principal diagonal or leading diagonal.
• Diagonal Matrix: A square matrix \( [a_{ij}] \) is said to be a diagonal matrix if \( a_{ij} = 0 \) for \( i \neq j \).

  In other words, a square matrix is said to be a diagonal matrix, if its element not on principal diagonal are zero.

• Scalar Matrix: A square matrix \( A = [a_{ij}]_{n \times n} \) is called a scalar matrix, if

  (i) \( a_{ij} = 0 \forall i \neq j \) and (ii) \( a_{ii} = c \forall i \), where \( c \neq 0 \).

  In other words, a square matrix is said to be scalar, if it is a diagonal matrix and entries on its principal diagonal are equal.

• Identity Matrix: A square matrix in which all non diagonal elements are zero and all diagonal elements are equal to 1 is called identity matrix.

  i.e., \( I = [a_{ij}]_{n \times n} \) is an identity matrix if

  \( a_{ij} = 0 \forall i \neq j \) and \( a_{ij} = 1 \forall i = j \)

  For example, \( \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}_{3 \times 3} \) is an identity matrix.

• Null or Zero Matrix: A matrix whose all elements are zero is called a null matrix or a zero matrix i.e., \( A = [a_{ij}]_{m \times n} \) is null matrix if \( a_{ij} = 0, \forall i, j \).
• Upper and Lower Triangular Matrices: A square matrix \( A = [a_{ij}] \) is called

  (i) an upper triangular matrix, if \( a_{ij} = 0 \forall i > j \), i.e., all entries below principal diagonal are zero.

  (ii) a lower triangular matrix, if \( a_{ij} = 0 \forall i < j \), i.e., all entries above principal diagonal are zero.

• Equality of Matrices: Two matrices \( A = [a_{ij}]_{m \times n} \) and \( B = [b_{ij}]_{m \times n} \) of the same order are equal, if

  \( a_{ij} = b_{ij} \forall i = 1, 2, ..., m \) and \( j = 1, 2, ..., n \).

• Addition of Matrices: If \( A = [a_{ij}]_{m \times n} \) and \( B = [b_{ij}]_{m \times n} \) are two matrices of the same order \( m \times n \), then their sum \( A + B \) is an \( m \times n \) matrix such that

  \( (A + B)_{ij} = a_{ij} + b_{ij} \forall i = 1, 2, ..., m \) and \( j = 1, 2, 3, ..., n \)

  Following are the properties of matrix addition:

  (i) Commutativity: If \( A \) and \( B \) are two matrices of the same order, then
  \( A + B = B + A \)

  (ii) Associativity: If \( A, B \) and \( C \) are three matrices of the same order, then
  \( (A + B) + C = A + (B + C) \)

  (iii) Existence of Identity: The null matrix is the identity element for matrix addition i.e.,
  \( A + O = A + O = A \)

  (iv) Existence of Inverse: For every matrix \( A = [a_{ij}]_{m \times n} \) there exists a matrix \( - A = [- a_{ij}]_{m \times n} \) such that
  \( A + (- A) = O = (- A) + A \)

  (v) Cancellation Laws: If \( A, B \) and \( C \) are three matrices of the same order, then
  \( A + B = A + C \)
  \( \implies \) \( B = C \) and \( B + A = C + A \)
  \( \implies \) \( B = C \)

• Scalar Multiplication: Let \( A = [a_{ij}] \) be an \( m \times n \) matrix and \( k \) be any number called a scalar. Then, the matrix obtained by multiplying every element of \( A \) by \( k \) is called the scalar multiple of \( A \) by \( k \) and is denoted by \( kA \).

  Thus, \( kA = [ka_{ij}]_{m \times n} \)

  Following are the properties of scalar multiplication:

  If \( A \) and \( B \) are two matrices of the same order and \( k, l \) are scalars, then

  (i) \( k(A + B) = kA + kB \)

  (ii) \( (k + l) A = kA + lA \)

  (iii) \( (kl) A = k(lA) = l(kA) \)

  (iv) \( (-k) A = - (kA) = k(- A) \)

  (v) \( 1 A = A \)

  (vi) \( (-1) A = - A \)

  Note that a scalar matrix can be obtained by multiplying an identity matrix by a scalar.

• Subtraction of Matrices: If \( A \) and \( B \) are two matrices of the same order, then \( A - B = A + (- B) \).
• Multiplication of Matrices: Two matrices \( A \) and \( B \) are said to be defined for multiplication, if the number of columns of \( A \) (pre multiplier) is equal to the number of rows of \( B \) (post-multiplier).

  For example, if the order of \( A \) (pre-multiplier) is \( m \times n \) and the order of \( B \) (post-multiplier) is \( n \times p \) then \( A \) and \( B \) is defined for multiplication and order of product of \( A \) and \( B \) denoted by \( AB \) is \( m \times p \).

  i.e., \( A_{m \times n} \times B_{n \times p} = AB_{m \times p} \)

  Definition of Product : Let \( A = [a_{ij}]_{m \times n} \) and \( B = [b_{jk}]_{n \times p} \) be two matrices then product of \( A \) and \( B \) denoted by \( AB \) is given as

  \( AB = [c_{ij}]_{m \times p} \)

  where, \( C_{ij} = a_{i1}b_{1j} + a_{i2}b_{2j} + ... + a_{in}b_{nj} = \sum_{r=1}^{n} a_{ir}b_{rj} \) [\( 1 \le i \le m \) and \( 1 \le j \le p \)]

  Here, \( A \) is pre-multiplier or pre-factor \( B \) is post-multiplier or post-factor.

  The diagram given below may help the students to understand the process of finding the product of two matrix:

  \( \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix}_{2 \times 3} \times \begin{bmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ b_{31} & b_{32} & b_{33} \end{bmatrix}_{3 \times 3} = \begin{bmatrix} C_{11} & C_{12} & C_{13} \\ C_{21} & C_{22} & C_{23} \end{bmatrix}_{2 \times 3} \)

  \( = C_{13} \)
  \( a_{11} \times b_{13} + a_{12} \times b_{23} + a_{13} \times b_{33} \)

  \( = C_{12} \)
  \( a_{11} \times b_{12} + a_{12} \times b_{22} + a_{13} \times b_{32} \)

  \( = C_{11} \)
  \( a_{11} \times b_{11} + a_{12} \times b_{21} + a_{13} \times b_{31} \)

  \( a_{21} \times b_{11} + a_{22} \times b_{21} + a_{23} \times b_{31} \)
  \( = C_{21} \)

  \( a_{21} \times b_{12} + a_{22} \times b_{22} + a_{23} \times b_{32} \)
  \( = C_{22} \)

  \( a_{21} \times b_{13} + a_{22} \times b_{23} + a_{23} \times b_{33} \)
  \( = C_{23} \)

  Matrix multiplication has the following properties:

  (i) Matrix multiplication is not commutative.

  (ii) Matrix multiplication is associative i.e., \( (AB) C = A (BC) \) wherever both sides of the equality are defined.

  (iii) Matrix multiplication is distributive over matrix addition i.e., \( A (B + C) = AB + AC \) and \( (B + C) A = BA + CA \) wherever both sides of the equality are defined.

  (iv) If \( A \) is an \( m \times n \) matrix, then \( I_m A = A = A I_n \)

  (v) If \( A \) is an \( m \times n \) matrix and \( O \) is a null matrix, then \( A_{m \times n} \times O_{n \times p} = O_{m \times p} \) and \( O_{p \times m} \times A_{m \times n} = O_{p \times n} \) i.e., the product of a matrix with a null matrix is a null matrix.

  (vi) In matrix multiplication the product of two non-zero matrices may be a 'zero-matrix' i.e., \( AB = 0 \), does not imply that at least one of the \( A \) or \( B \) should be zero.

• If \( A \) is a square matrix, then we define \( A^1 = A \) and \( A^{n+1} = A^n \cdot A \).
• If \( A \) is a square matrix and \( a_0, a_1, ..., a_n \) are constants, then

  \( a_0 A^n + a_1 A^{n-1} + a_2 A^{n-2} + ... + a_{n-1} A + a_n \) is called a matrix polynomial.

• Transpose of a Matrix: Let \( A = [a_{ij}] \) be an \( m \times n \) matrix. Then, the transpose of \( A \), denoted by \( A^T \), is an \( n \times m \) matrix such that

  \( (A^T)_{ij} = a_{ji} \forall i = 1, 2, ..., n; j = 1, 2, ..., m \)

  i.e., the matrix obtained by interchanging rows into columns, of a given matrix \( A \) is called the transpose of \( A \) and is denoted by \( A^T \) or \( A' \).

  Following are the properties of transpose of a matrix:

  (i) \( (A^T)^T = A \)

  (ii) \( (A + B)^T = A^T + B^T \)

  (iii) \( (kA)^T = k A^T \)

  (iv) \( (AB)^T = B^T A^T \)

  (v) \( (ABC)^T = C^T B^T A^T \)

• A square matrix \( A = [a_{ij}] \) is called a symmetric matrix, if

  \( a_{ij} = a_{ji} \forall i, j \) i.e., \( A = A^T \)

• A square matrix \( A = [a_{ij}] \) is called a skew symmetric matrix, if

  \( a_{ij} = - a_{ji} \forall i, j \) i.e., \( A^T = - A \)

• All main diagonal elements of a skew-symmetric matrix are zero.
• Every square matrix can be uniquely expressed as the sum of a symmetric and a skew-symmetric matrix.
• All positive integral powers of a symmetric matrix are symmetric.
• All odd positive integral powers of a skew-symmetric matrix are skew-symmetric.