The rectangular arrangement of an array of numbers is called a matrix. Matrices are the plural form of the matrix. These matrices are classified into various types depending on the number of elements present in them, number of rows, columns, order, size and so on. Some of the matrices types are row matrix, column matrix, null matrix, identity matrix, singleton matrix, and others.
Here we are giving the example questions and answers of classification of matrices in the following sections. Students can check those practice questions and solve them to get a good score in the exam.
Also, check
- Definition of a Matrix
- Classification of Matrices
- Position of an Element in a Matrix
- Oder of a Matrix
Solved Problems on Classification of Matrices
Problem 1:
Let \( A =\left[
\begin{matrix}
3 & 4 & 9\cr
0 & 1 & 3\cr
\end{matrix}
\right]
\), \( B =\left[
\begin{matrix}
0 & 0 & 0\cr
0 & 0 & 0\cr
0 & 0 & 0\cr
\end{matrix}
\right]
\), \( C =\left[
\begin{matrix}
8 & 9 & 5\cr
6 & 0 & 4\cr
1 & 5 & 2\cr
\end{matrix}
\right]
\), \( D =\left[
\begin{matrix}
14 & 8 & 5\cr
\end{matrix}
\right]
\), \( E =\left[
\begin{matrix}
100\cr
\end{matrix}
\right]
\)
Identify the type of each matrix.
Solution:
\( A =\left[
\begin{matrix}
3 & 4 & 9\cr
0 & 1 & 3\cr
\end{matrix}
\right]
\)
Matrix A is a rectangular matrix. As the number of rows is 2 which is not equal to the number of columns 3.
\( B =\left[
\begin{matrix}
0 & 0 & 0\cr
0 & 0 & 0\cr
0 & 0 & 0\cr
\end{matrix}
\right]
\)
Matrix B is a Null matrix as the all its elements are zero’s.
\( C =\left[
\begin{matrix}
8 & 9 & 5\cr
6 & 0 & 4\cr
1 & 5 & 2\cr
\end{matrix}
\right]
\)
Matrix C is a square matrix. As the number of rows = number of columns i.e 3 = 3.
\( D =\left[
\begin{matrix}
14 & 8 & 5\cr
\end{matrix}
\right]
\)
Matrix D is a row matrix as it has only one row.
\( E =\left[
\begin{matrix}
100\cr
\end{matrix}
\right]
\)
Matrix E is a singleton matrix as it has only one element.
Problem 2:
(i) Construct a 2 x 3 matrix that has elements as natural numbers from 10 to 16.
(ii) Construct a 2 x 2 simple identity matrix.
Solution:
(i) Matrix that has numbers from 10 to 16 is \( \left[
\begin{matrix}
10 & 11 & 12\cr
13 & 14 & 15\cr
\end{matrix}
\right]
\)
(ii) 2 x 2 Identity matrix is \( \left[
\begin{matrix}
1 & 0\cr
0 & 1\cr
\end{matrix}
\right]
\)
Problem 3:
Identify which of the following is a diagonal matrix.
\( A =\left[
\begin{matrix}
3 & 4 & 9\cr
12 & 11 & 35\cr
\end{matrix}
\right]
\), \( B =\left[
\begin{matrix}
5 & 0 & 0\cr
0 & 5 & 0\cr
\end{matrix}
\right]
\), \( C =\left[
\begin{matrix}
3 & 0\cr
0 & 1\cr
\end{matrix}
\right]
\)
Solution:
\( C =\left[
\begin{matrix}
3 & 0\cr
0 & 1\cr
\end{matrix}
\right]
\) is a diagonal matrix as its diagonal elements are non-zero and the remaining elements are zeros.
The remaining matrices don’t satisfy the diagonal matrix condition.
Problem 4:
\( A =\left[
\begin{matrix}
1 & 6 & 7\cr
2 & 4 & 6\cr
3 & 2 & 5\cr
\end{matrix}
\right]
\), \( B =\left[
\begin{matrix}
0 & 0 & 0\cr
0 & 0 & 0\cr
0 & 0 & 0\cr
\end{matrix}
\right]
\), \( C =\left[
\begin{matrix}
18 & 0\cr
0 & 18\cr
\end{matrix}
\right]
\), \( D =\left[
\begin{matrix}
12 & 10 & 0 & 2\cr
8 & 5 & 4 & 6\cr
\end{matrix}
\right]
\), \( E =\left[
\begin{matrix}
4 & 0 & 0\cr
2 & 5 & 0\cr
1 & 3 & 6\cr
\end{matrix}
\right]
\), \( F =\left[
\begin{matrix}
180 & 0 & 0\cr
0 & 120 & 0\cr
0 & 0 & 100\cr
\end{matrix}
\right]
\)
Identify the following matrices.
(i) Horizontal matrix
(ii) Scalar matrix
(iii) Null matrix
(iv) Square Matrix
(v) Triangular Matrix
(vi) Diagonal Matrix
Solution:
(i) D is the horizontal matrix as its number of rows are less than the number of columns.
(ii) C is a scalar matrix. Here the diagonal elements are the same and the remaining elements are zeros.
(iii) B is a null matrix as all its elements are zeros.
(iv) A is a square matrix
(v) E is a lower triangular matrix
(vi) F is a diagonal matrix.
Problem 5:
Give a 4 x 4 order example for the below-mentioned matrices.
(i) Symmetric Matrix
(ii) Upper Triangular Matrix
(iii) Boolean Matrix
(iv) Matrix of Ones
Solution:
(i) \( X =\left[
\begin{matrix}
3 & 6 & 9 & 12\cr
6 & 12 & 18 & 24\cr
9 & 18 & 27 & 36\cr
12 & 24 & 36 & 42\cr
\end{matrix}
\right]
\) is a symmetric matrix.
(ii) \( Y =\left[
\begin{matrix}
1 & 2 & 3 & 5\cr
0 & 4 & 6 & 7\cr
0 & 0 & 8 & 9\cr
0 & 0 & 0 & 11\cr
\end{matrix}
\right]
\) is an upper triangular matrix.
(iii) \( Z =\left[
\begin{matrix}
1 & 0 & 0 & 1\cr
0 & 1 & 1 & 0\cr
1 & 0 & 0 & 1\cr
0 & 0 & 1 & 1\cr
\end{matrix}
\right]
\)
(iv) \( S =\left[
\begin{matrix}
1 & 1 & 1 & 1\cr
1 & 1 & 1 & 1\cr
1 & 1 & 1 & 1\cr
1 & 1 & 1 & 1\cr
\end{matrix}
\right]
\) is a matrix of ones.