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Matrices

Types of Matrices

Matrices are distinguished on the basis of their order, elements and certain other conditions. There are different types of matrices but the most commonly used are discussed below. Let’s find out the types of matrices in the field of mathematics.

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Types of Matrices

Different types of Matrices and their forms are used for solving numerous problems. Some of them are as follows:

Types of Matrices

1) Row Matrix

A row matrix has only one row but any number of columns. A matrix is said to be a row matrix if it has only one row. For example, $$ A =\begin{bmatrix} -1/2 & √5 & 2 & 3\end{bmatrix}$$ is a row matrix of order 1 × 4. In general, A = [aij]1 × n is a row matrix of order 1 × n.

2) Column Matrix

A column matrix has only one column but any number of rows. A matrix is said to be a column matrix if it has only one column. For example, $$ A =\begin{bmatrix} 0\\ √3\\-1 \\1/2 \end{bmatrix}$$ is a column matrix of order 4 × 1. In general, B = [bij]m × 1 is a column matrix of order m × 1.

3) Square Matrix

A square matrix has the number of columns equal to the number of rows. A matrix in which the number of rows is equal to the number of columns is said to be a square matrix. Thus an m × n matrix is said to be a square matrix if m = n and is known as a square matrix of order ‘n’. For example, $$ A =\begin{bmatrix} 3 & -1 & 0\\ 3/2 & √3/2 & 1\\4 & 3 & -1\end{bmatrix}$$ is a square matrix of order 3. In general, A = [aij] m × m is a square matrix of order m.

matrix cheat sheet

4) Rectangular Matrix

A matrix is said to be a rectangular matrix if the number of rows is not equal to the number of columns. For example, $$ A =\begin{bmatrix} 3 & -1 & 0\\ 3/2 & √3/2 & 1\\4 & 3 & -1\\ 7/2 & 2 & -5 \end{bmatrix}$$ is a matrix of the order 4 × 3

5) Diagonal matrix

A square matrix B = [bij] m × m is said to be a diagonal matrix if all its non-diagonal elements are zero, that is a matrix B =[bij]m×m is said to be a diagonal matrix if bij = 0, when i ≠ j. For example, $$ A =\begin{bmatrix} 4\end{bmatrix}\begin{bmatrix} -1 & 0\\ 0 & 2 \end{bmatrix}\begin{bmatrix} 3 & 0 & 0\\ 0 & -5 & 0\\ 0 & 0 & 2 \end{bmatrix}$$ are diagonal matrices of order 1, 2, 3, respectively.

6) Scalar Matrix

A diagonal matrix is said to be a scalar matrix if all the elements in its principal diagonal are equal to some non-zero constant. 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 scalar matrix if

  • bij = 0, when i ≠ j
  • bij = k, when i = j, for some constant k.

For example, $$ A =\begin{bmatrix} 4\end{bmatrix}\begin{bmatrix} -1 & 0\\ 0 & -1 \end{bmatrix}\begin{bmatrix} 3 & 0 & 0\\ 0 & 3 & 0\\ 0 & 0 & 3 \end{bmatrix}$$ are scalar matrices of order 1, 2 and 3, respectively.

7) Zero or Null Matrix

A matrix is said to be zero matrix or null matrix if all its elements are zero.
For Example, $$ A =\begin{bmatrix} 0\end{bmatrix}\begin{bmatrix} 0 & 0\\ 0 & 0 \end{bmatrix}\begin{bmatrix} 0 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{bmatrix}$$are all zero matrices of the order 1, 2 and 3 respectively. We denote zero matrix by O.

8) Unit or Identity Matrix

If a square matrix has all elements 0 and each diagonal elements are non-zero, it is called identity matrix and denoted by I.
Equal Matrices: Two matrices are said to be equal if they are of the same order and if their corresponding elements are equal to the square matrix A = [aij]n × n is an identity matrix if

  • aij = 1 if i = j
  • aij = 0 if i ≠ j

We denote the identity matrix of order n by In. When the order is clear from the context, we simply write it as I. For example, $$ A =\begin{bmatrix} 1\end{bmatrix}\begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix}$$ are identity matrices of order 1, 2 and 3, respectively. Observe that a scalar matrix is an identity matrix when k = 1. But every identity matrix is clearly a scalar matrix.

9) Upper Triangular Matrix

A square matrix in which all the elements below the diagonal are zero is known as the upper triangular matrix. For example, $$ A =\begin{bmatrix} 3 & -5 & 7\\ 0 & 4 & 0\\ 0 & 0 & 9 \end{bmatrix}$$

10) Lower Triangular Matrix

A square matrix in which all the elements above the diagonal are zero is known as the upper triangular matrix. For example, $$ A =\begin{bmatrix} 3 & 0 & 0\\ 0 & 4 & 0\\ -5 & 7 & 9 \end{bmatrix}$$

Solved Examples For You

Question 1: Assertion :  $$ A =\begin{bmatrix} 3 & 0 & 0\\ 0 & 4 & 0\\ 0 & 0 & 7 \end{bmatrix}$$ is a diagonal matrix.

Reason: If A=[aijis a square matrix such that aij=0,ijthen is called a diagonal matrix.
  1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
  2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
  3. Assertion is correct but Reason is incorrect
  4. Both Assertion and Reason are incorrect

Answer : If A=[aij]n×n is a square matrix such that aij 0 for ij, then is called a diagonal matrix. Since, a12 a13 a21 a23 a31 a32 Thus, the given statement  is true and $$ A =\begin{bmatrix} 3 & 0 & 0\\ 0 & 4 & 0\\ 0 & 0 & 7 \end{bmatrix}$$ is a diagonal matrix is a diagonal matrix

Question 2: What is meant by matrices and what are its types?

Answer: Matrix refers to a rectangular array of numbers. A matrix consists of rows and columns. These rows and columns define the size or dimension of a matrix. The various types of matrices are row matrix, column matrix, null matrix, square matrix, diagonal matrix, upper triangular matrix, lower triangular matrix, symmetric matrix, and antisymmetric matrix.

Question 3: Explain a scalar matrix?

Answer: The scalar matrix is similar to a square matrix. In a scalar matrix, all off-diagonal elements are equal to zero and all on-diagonal elements happen to be equal. In other words, we can say that a scalar matrix is an identity matrix’s multiple.

Question 4: Can we say that a zero matrix is invertible?

Answer: No, a zero matrix is not invertible. This is because its determinant is zero.

Question 5: What is meant by a symmetric matrix?

Answer: A symmetric matrix refers to a square matrix whose transpose is equal to it. Furthermore, it is possible only for square matrices to be symmetric because equal matrices have equal dimensions. A symmetric matrix has symmetric entries with respect to the main diagonal.

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