TYPES OF MATRICES

Definition: Matrix:

  • A set of mn elements arranged in a rectangular arrangement along m rows and n columns enclosed by the brackets [ ] or ( ) is called m by n matrix.
  • Example: matrix

 

Order of matrix

  • The order of a matrix is a number of rows and columns of the matrix.
  • Example: order of matrix
  • From above example the order of matrix ot has no of rows=2 and no of columns=4.

 

Row matrix

  • The Matrix has a single row is called a row matrix.
  • Example: row matrix

 

Column matrix

  • The matrix has a single column is called a column matrix.
  • Example: column of matrix

 

Square matrix

  • A matrix having an equal number of rows and columns is called a square matrix.
  • Example: square matrix

Null matrix

  • In a matrix, if all the elements are zero then that matrix is called a null or zero matrices and it is always denoted by 0.
  • Example: null matrix

 

Diagonal matrix

  • In a square matrix, all the elements except the elements in the main diagonal are zeros, then the matrix is called a diagonal matrix.
  • Example: diagonal matrix

Scalar matrix

  • A square matrix in which all the elements of its leading diagonal are equal and the other elements all the zero is called a scalar matrix.
  • Example scalar matrix

Unit matrix

  • A diagonal matrix that has unity in the leading diagonal is called a unit matrix. It is always denoted by I.
  • Example: unit matrix

Upper triangular matrix

  • A square matrix in which all elements below the leading diagonal zero is called an upper triangular matrix.
  • Example: upper triangular matrix

Lower triangular matrix

  • A square matrix in which all elements above the leading diagonal are zero is called a lower triangular matrix.
  • Example: lower triangular matrix

Transpose of a matrix

  • The matrix got from the given matrix A, by interchanging the rows and columns is called the transpose of A and denoted by A^T.
  • Example:transpose of matrix

Symmetric matrix

  • A square matrix A is symmetric if  A =A^T.
  • Example: symmetric matrix

Skew symmetric matrix

  • In skew- symmetric A square matrix, A then skew-symmetric is A=-A^T.
  • Example: skew symmetric matrix

Singular matrix

  • In a singular matrix, a square matrix A is said to be singular if the value of A is zero.
  • Example: singular matrix

Equality of matrices

  • Two matrices A and B is said to be equal if and only if
  1.  Both matrices of the same order
  2.  Each element A is equal to the corresponding element of B.
  3. Example: Equality of matrix

Addition of matrices

  • Let A and B be two matrices of same orders, then their sum A+B is defined as the matrix where each element is the sum of the corresponding elements of A and B.
  • Example: addition of matrix

Subtraction of matrices

  • Let A and B be two matrices of the same orders, then their difference A-B is defined as the matrix where each element is the difference of the corresponding elements of A and B.
  • Example: subtraction of matrix

Scalar Multiplication

  • The product of a scalar matrix A by a scalar K is a matrix in which each element is K times the corresponding elements of A.
  • Example:  k= 2, A=\begin{bmatrix} 1 &4 &2 \\ 7 & 0 &1 \\ 2&3 &-1 \end{bmatrix}\Rightarrow KA=2\begin{bmatrix} 1 &4 &2 \\ 7& 0 & 1\\ 2& 3 &-1 \end{bmatrix}\Rightarrow \begin{bmatrix} 1\times 2 & 4\times 2 &2\times 2 \\ 7\times 2& 0\times 2 &1\times 2 \\ 2\times 2& 3\times 2&-1\times 2 \end{bmatrix}\Rightarrow \begin{bmatrix} 2 &8 &4 \\ 14 &0 & 2\\ 4 & 6 &-2 \end{bmatrix}

Multiplication of matrices

  • Two matrices can be multiplied only when the number of columns in the first matrix is equal to the number of rows in the second matrix.
  • E.g., if A = m×n, B = n×p then AB = m×p
  • Example: A=\begin{bmatrix} 3 & 1\\ 6 & 4 \end{bmatrix};B=\begin{bmatrix} 2& 7 &10 \\ 5 & 8 & 9 \end{bmatrix}\Rightarrow A\times B=\begin{bmatrix} 3\times 2+1\times 5 &3\times 7+1\times 8 &3\times 10+1\times 9 \\ 6\times 2+4\times 5 &6\times 7+4\times 8 & 6\times 10+4\times 9 \end{bmatrix}\Rightarrow \begin{bmatrix} 6+5 &21+8 &30+9 \\ 12+20 &42+32 & 60+36 \end{bmatrix}\Rightarrow \begin{bmatrix} 11 &29 &39 \\ 32&74 &96 \end{bmatrix}

 

 

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