Question

What are special matrices?

Answer 1

Matrix consists of entries in rows and columns. Special matrices have unique features and those features distinguish each of the special matrices. Matrix with the same number of rows and columns is a square matrix. Then, we have the triangular matrix, which can be an upper or lower triangular matrix. Triangular matrix which has zeroes below the main diagonal is the upper triangular matrix and the one with zeroes above the main diagonal is the lower triangular matrix. Symmetric matrix remains the same even when transposed.

Complete step-by-step solution:
According to the question given to us, we have to write about special matrices.
Matrix is a set of entries enclosed within brackets. It consists of rows and columns and so the size of these matrices are denoted with the help of the number of rows and columns.
For example - $\left( \begin{matrix} 1 & 2 \\\ 4 & 3 \\\ \end{matrix} \right)$ is a matrix and its size is $2\times 2$ as the matrix has two rows and two columns respectively.
The first special matrix is a square matrix.
Square matrix can be said to be a matrix with the same number of rows and columns. Examples of square matrices have the sizes as: $2\times 2$, $3\times 3$, $4\times 4$, etc.

1 & 2 \\\ 4 & 3 \\\ \end{matrix} \right)$$ is a square matrix of the order $$2\times 2$$. Here, the main diagonal elements are 1 and 3. $$\left( \begin{matrix} 2 & 3 & 2 \\\ 1 & 2 & 3 \\\ 3 & 5 & 4 \\\ \end{matrix} \right)$$ is a square matrix of the order $$3\times 3$$. Then, we have triangular matrices. For a triangular matrix, we need a square matrix as a base. Triangular matrices are of two types: 1) Upper triangular matrix – The upper triangular matrix has all entries zero below the main diagonal. For example – $$\left( \begin{matrix} 1 & 2 \\\ 0 & 3 \\\ \end{matrix} \right)$$ is an upper triangular matrix with an order $$2\times 2$$. $$\left( \begin{matrix} 2 & 3 & 2 \\\ 0 & 2 & 3 \\\ 0 & 0 & 4 \\\ \end{matrix} \right)$$ is an upper triangular matrix with the order $$3\times 3$$. 2) Lower triangular matrix – The lower triangular matrix has all entries as zero above the main diagonal. For example – $$\left( \begin{matrix} 1 & 0 \\\ 4 & 3 \\\ \end{matrix} \right)$$ is a lower triangular matrix of the order $$2\times 2$$. $$\left( \begin{matrix} 2 & 0 & 0 \\\ 4 & 2 & 0 \\\ 1 & 6 & 4 \\\ \end{matrix} \right)$$ is a lower triangular matrix of the order $$3\times 3$$. Next special matrix we have is, symmetric matrix. A symmetric matrix is a matrix which even when transposed gives the same matrix. And it should be a square matrix. Transpose is an operation on the matrix which causes the rows elements to become column elements and the column elements to row elements. For example – If $$A=\left( \begin{matrix} 1 & 0 \\\ 4 & 3 \\\ \end{matrix} \right)$$, Then the transpose of A is, $${{A}^{T}}=\left( \begin{matrix} 1 & 4 \\\ 0 & 3 \\\ \end{matrix} \right)$$ So, a symmetric matrix remains the same even when they are transposed. For example – $$A=\left( \begin{matrix} 1 & 2 \\\ 2 & 3 \\\ \end{matrix} \right)$$, $$\left( \begin{matrix} 2 & 0 & 1 \\\ 0 & 2 & 6 \\\ 1 & 6 & 4 \\\ \end{matrix} \right)$$ And a symmetric matrix with only the diagonal elements is a diagonal matrix. For example – $$A=\left( \begin{matrix} 1 & 0 \\\ 0 & 3 \\\ \end{matrix} \right)$$, $$\left( \begin{matrix} 2 & 0 & 0 \\\ 0 & 2 & 0 \\\ 0 & 0 & 4 \\\ \end{matrix} \right)$$ **Note:** We can see that all the special matrices that we came across have a square matrix as its base. So, it is helpful in recognizing a special matrix. Also, don’t get confused with the different special matrices as there is little difference between each of them, but that makes the whole of the difference.