What is Matrix?

Matrix is simply a multidimensional array.

  • If m != n, Rectanguar Matrix

  • If m == n, Square Matrix

    • Unit Matrix

      A Square Matrix whose all Principal Diagonal matrix are equal to 1, and remaing elements are 0.

    • Null Matrix

      A Square or Non-Square (rectangular) matrix with all elements equal to 0 is termed as Null Or Zero matrix.

    • Row Matrix

      Matrix having only 1 Row.

    • Column Matrix

      Matrix having only 1 Column.

    • Sub-Matrix

      Any matrix obtained by omitting some rows and Columns from a given, (m X n) matrix ‘A’, is called Sub-matrix of ‘A’.

      • Principal Sub-Matrix: A Square Sub-matrix of a Square matrix ‘A’ is called Principal Sub-matrix, if its (sub-matrix) diagonal elements are also the diagonal elements of matrix ‘A’.

Equality of Matrix

Two or more matrices are said to be Equal if they are of:

  1. Same Size, order should be same.
  2. The Elements in the corresponding places are same.

Types of Square Matrix

  • Upper Triangular Matrix

    A Square matrix, A = [a i j ] is called an upper triangular matrix if a[ i ][ j ] = 0, whenever i > j.

  • Lower Triangular Matrix

    A Square matrix, A = [a i j ] is called a lower triangular matrix if a[ i ][ j ] = 0, whenever i < j.

    • A Triangular matrix A = [a i j ] m x n is called Strictly triangular matrix if a[ i ][ i ] = 0, for i = 1, 2, …, n.
  • Diagonal Matrix

    A Square matrix, A = [a i j ]n x n whose elements above and below the principal diagonal are all zero.

  • Scalar Matrix

    A diagonal matrix whose diagonal elements are all equal is called a Scalar matrix.

Addition of Matrices

Addition of Matrices is only possible when matrices are Conformable, i.e Same Size.

  • Properties of Addition
    1. Commutative: A + B = B + A
    2. Associative: (A + B) + C = A + (B + C)
    3. Existence of additive identity : A + 0 = 0 + A
    4. Existence of additive inverse : -A + A = 0 = A + (-A)
    5. Cancellation law:
    • A + B = A + C -> B = C
    • B + A = C + A -> B = C

Subtraction of Matrices

Subtaction of Matrices is only possible when matrices are Conformable, i.e Same Size. A - B = A + (-B)

Properties of addition also holds in Subtraction.

Scalar multiplication of Matrix

If A = [a i j ]m x n, then

  • K * A = A * K = [K a i j]m x n

  • Properties of Scalar Multiplication
    1. K * (A + B) = KA + KB
    2. (P + Q) * A = PA + QA
    3. P * (Q * A) = (P * Q) * A
    4. (-K) * A = -(K * A) = K * (-A)

Multiplication of Matrices

Multiplication of matrices is only possible when they are multiplication conformable, that is - number of column in first matrix should be equal to number of rows in second matrix.

  • Each element of ith row in first matrix is multiplied with corressponding element of ith column in second matrix, and the summation of each multiplication is the ith element of the resultant matrix.

  • Properties of matrix multiplication
    1. Matrix multiplication is Associative (A * B) * C = A * (B * C)
    2. Matrix multiplication is distributive with respect to addition of matrices: A * (B + C) = AB + AC
    3. Matrix Multiplication is not Always Commutative: AB != BA
    • If AB == BA, then the matrices ‘A’ and ‘B’ are said to commute.
    • If AB == -BA, then the matrices are said to be anti-commute.
      1. The equation AB = 0, does not necessarily imply that one of the matrices ‘A’ and ‘B’ must be a Zero matrix.
    • If AB = 0, then it does not necessarily imply that BA = 0.
      1. If A is an m x n matrix, In denotes the n-rowed unit matrix, it can be easily seen that: A * In = A = Im * A

Echelon Form

A matrix A of order m x n is said to be in row echelon form if:

  1. Zero rows (If any occur) then they must be below the non-zero rows.
  2. The number of Zeros before the first non-zero elements in each row is less than the number of such zeros in the next non-zero row.

If a Matrix is in Row Echelon form then, Rank of matrix = Number of Non-Zero Rows.

Key Points

  1. If ‘A’ is a square matrix of size(n x n), then r(A) <= n.
  2. If ‘A’ is a non-square matrix of size (m x n), then r(A) <= min(m, n).
  3. If a matrix a attains the maximum possible rank, then it is said to have Full Rank.
  4. If a square matrix is non-singular, then it will have full rank.
  5. Rank of a matrix = The number of linearly independent rows == The number of linearly independent columns of the matrix.
  6. If all the rows (or Columns) of a matrix are identical or Proportional to each other, then its Rank = 1.