Revision Notes: Matrices | Mathematics Class 10 ICSE PDF Download

Important Terms

• A set of mn numbers arranged in the form of a rectangular array of m rows and n columns is called an m x n matrix
• Each number or entity in a matrix is called its element.
• In a matrix, the horizontal lines are called rows, whereas the vertical lines are called columns.
• If a matrix contains m rows and n columns then it is said to be a matrix of order m x 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 having only one row is known as a row matrix.
• A matrix having only one column is known as a column matrix.
• A matrix which has an equal number of rows and columns is called a square matrix.
• Rectangular matrix: A matrix in which number of rows are 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 a null matrix.
• A square matrix which has every non–diagonal element as zero is called a diagonal matrix.
• A diagonal matrix in which each element of its leading diagonal is unity is called identity matrix.
• Two matrices are said to be equal, if they are of the same order and have the same corresponding elements.
• If A is a matrix, then its transpose is obtained by interchanging its rows and columns. Transpose of a matrix A is denoted by A^t.

Operations on matrices

• Addition of Matrices: Let A and B be two matrices each of order m x n. Then their sum A + B is a matrix of order m  n and is obtained by adding the corresponding elements of A and B.
• Subtraction of Matrices: Let A and B be two matrices each of order m x n. Then their difference A – B is a matrix of order m  n and is obtained by subtracting the corresponding elements of A and B.
• In addition or subtraction of the matrices, the order of the resulting matrix is the same as the order of matrices added or subtracted.
• Matrix addition is commutative
  i.e., A + B = B + A
• Matrix addition is associative for any three matrices A, B and C.
  A + (B + C) = (A + B) + C.
• If A and B are two matrices, A + X = B ⇒ X = B - A
• Additive Identity: If any matrix is added to null (zero) matrix of the same order, or a null matrix is added to a matrix of the same order, the matrix remains unaltered and hence, the null matrix is said to be the additive identity in matrices.
  A null matrix is identity element for addition
  i.e., A + 0 = A = 0 + A.
• If A and B are two matrices of the same order such that:
  A + B = B + A = a null matrix,
  then A is said to be the additive inverse of B and B is said to be the additive inverse of A. 
  Additive inverse of a matrix A is its negative A.
• If O is the null or zero matrix of the same order as matrix A, then
  A + (–A) = (–A) + A = O
• Let A and B are two matrices of the same order such that 
  A + X = B, where X is an unknown matrix; then X = B – A and the order of matrix X is same as that of A and B.
• The multiplication of a matrix A by a number k gives a matrix of the same order as A, in which all the elements are k times the elements of A.
• The product of two matrices A and B is defined if the number of columns of A is equal to the number of rows of B.
• Let A = [a_ij] be an m × n matrix and B = [b_jk] 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 c_ik of the matrix C, we take the i^th row of A and k^th column of B, multiply them element wise and take the sum of all these products.
• Matrix multiplication is not commutative. In general, AB ≠ BA.
• The product of two non-zero matrices can be a zero matrix.
• Let A, B and C are matrices.
  Then AB = AC, A ≠ 0 ⇒ it is not necessary that B = C.
  In general, cancellation law is not applicable in matrix multiplication.
• Identity matrix: The unit matrix I is known as the identity matrix for multiplication.
  Let A be any square matrix and I be the unit matrix of same order, then,
  A x I = I x A = A.
• Let A, B and C be any three matrices.
  Then (AB) C = A (BC)
  Thus, matrix multiplication is associative.
• Let A and B are two matrices.
  Then AB = 0 ⇒ it is not necessary that A = 0 or B = 0
• Let A and B are two matrices.
  Then If A = 0 or B = 0, then AB = 0 = BA
• Let A, B and C be any three matrices.
  1. then A(B + C) = AB + AC and A (B - C) = AB - AC
  2. then (A + B) C = AC + BC and (A - B)C = AC - BC
     In general, matrix multiplication is distributive over addition and subtraction.
• Laws of algebra are not applicable to matrices.
  That is,
  (A + B)² ≠ A² + 2AB + B²
  and
  (A + B)(A - B) ≠ A² - B²