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Types Of Matrices

    In the previous article, we learned about systems of linear equations can be represented using a matrix or augmented matrix. There are many types of matrices which we are going to explore in this post.

    In general matrix are referred using their order which is m \times n where m is the rows and n is number of columns in the matrix.

    If A[a_{ij}] is a matrix of order m \times n , then, 

    A = [a_{ij}]_{m \times n} = \begin{bmatrix} a_{11} & a_{12} & a_{1n}\\ a_{21} & a_{22} & a_{2n}\\ a_{m1} & a_{m2} & a_{mn}\end{bmatrix}

    Types of Matrices

    There are many types of matrix in linear systems. We have listed few important ones.

    • Square matrix
    • Diagonal matrix
    • Scalar matrix
    • Unit or Identity matrix
    • Null matrix
    • Upper triangular
    • Lower triangular matrix

    There are some other types about which we shall discuss later. Let us try to know these basic matrices more.

    Square Matrix

    A matrix of order m \times n where m = n is known as square matrix. For example,

    A = \begin{bmatrix}1 & 4 & 9\\ 3 & 7 & 3 \\ 1 & 3 & 1\end{bmatrix}_{3 \times 3}

    Note: a_{11}, a_{22}, a_{33} are diagonal elements.

    Diagonal Matrix

    A square matrix with all diagonal elements as 0 is called a diagonal matrix, but the diagonal elements may or may not be zero. For example,

    A = \begin{bmatrix}1 & 0 & 0\\ 0 & 7 & 0 \\ 0 & 0 & 1\end{bmatrix}_{3 \times 3}

    Scalar Matrix

    A diagonal matrix with equal diagonal elements are called a scalar matrix. The scalar matrix is obtained by multiplying the identity matrix with a scalar value. 

    \begin{aligned}
&A = \begin{bmatrix}7 & 0 & 0\\ 0 & 7 & 0 \\ 0 & 0 & 7\end{bmatrix}_{3 \times 3}\\\\
&B= \begin{bmatrix}2 & 0\\ 0 & 2\end{bmatrix}_{2 \times 2}
\end{aligned}

    Unit Matrix or Identity Matrix

    A square matrix with diagonal elements as 1 and all non-diagonal elements as 0 is known as a Unit or an Identity matrix. Also, note that the unit matrix is a scalar matrix in itself.

    \begin{aligned}
&I_3 = \begin{bmatrix}1 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}_{3 \times 3}\\\\
&I_2 = \begin{bmatrix}1 & 0\\ 0 & 1\end{bmatrix}_{2 \times 2}
\end{aligned}

    Null Matrix

    Usually no one creates null matrix, it is obtained due to some algebraic operations performed in matrices. A m \times n matrix with all elements equal to zero is called a null matrix.

    \begin{aligned}
&O_3 = \begin{bmatrix}0 & 0 & 0\\ 0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}_{3 \times 3}\\\\
&O_2 = \begin{bmatrix}0 & 0\\ 0 & 0\end{bmatrix}_{2 \times 2}
\end{aligned}

    Upper Triangular Matrix

    Matrix with lower diagonals equal to zero is called an Upper triangular matrix. For example,

    \begin{aligned}
&U_{3 \times 3} = \begin{bmatrix}1 & 2 & 3\\ 0 & 4 & 5 \\ 0 & 0 & 6\end{bmatrix}_{3 \times 3}
\end{aligned}

    For upper triangular matrix, a_{ij} = 0, for i > j and denoted by U.

    Lower Triangular Matrix

    The lower triangular matrix has its upper diagonals as zero and it is denoted by L. 

    \begin{aligned}
L_{3 \times 3} = \begin{bmatrix}1 & 0 & 0\\ 4 & 2 & 0 \\ 7 & 3 & 3\end{bmatrix}_{3 \times 3}
\end{aligned}

    For lower triangular matrix, a_{ij} = 0, for i < j and denoted by L.

    In the next post, we shall discuss basic row operations on matrices which are very useful in solving system of linear equations.

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