M
MATRICES & DETERMINANTS
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MATRIX
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Order of a matrix
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Types of Matrices
(i) Column matrix: A matrix is said to be a column matrix if it has only one column e.g. \(\begin{bmatrix}2\\3\end{bmatrix}\)
(ii) Row matrix: A matrix is said to be a row matrix if it has only one row e.g. \(\begin{bmatrix}1 & 2 & 3\end{bmatrix}\)
(iii) Square matrix: A matrix in which the number of rows are 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’ e.g. \(\begin{bmatrix}a & b & c\\ d & e & f\\ g & h & i\end{bmatrix}\)
(iv) 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
e.g. \(\begin{bmatrix}a & 0 & 0\\ 0 & b & 0\\ 0 & 0 & c\end{bmatrix}\)
(v) Scalar matrix: 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 and bij = k, when i = j, for some constant k.
e.g. \(\begin{bmatrix}a & 0 & 0\\ 0 & a & 0\\ 0 & 0 & a\end{bmatrix}\)
(vi) Identity matrix: A square matrix in which elements in the diagonal are all 1 and rest are all zero is called an identity matrix.
e.g. \(\begin{bmatrix}1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\end{bmatrix}\)
(vii) Zero matrix: A matrix is said to be zero matrix or null matrix if all its elements are zero. For example, [0].
e.g. \(\begin{bmatrix}0 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0\end{bmatrix}\)
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Addition and subtraction of matrices
Two matrices A and B can be added or subtracted if they are of the same order i.e. if A and B are two matrices of order m × n then A ± B is also a matrix of order m × n.
Example 1: Addition of Two Matrices
Let
$$
A=\begin{bmatrix}
2 & 3\\
4 & 5
\end{bmatrix},
\qquad
B=\begin{bmatrix}
1 & 6\\
7 & 2
\end{bmatrix}
$$
Then,
$$
A+B=
\begin{bmatrix}
2+1 & 3+6\\
4+7 & 5+2
\end{bmatrix}
$$$$=
\begin{bmatrix}
3 & 9\\
11 & 7
\end{bmatrix}
$$
Example 2: Subtraction of Two Matrices
Using the same matrices,
$$
A-B=
\begin{bmatrix}
2-1 & 3-6\\
4-7 & 5-2
\end{bmatrix}
$$$$=
\begin{bmatrix}
1 & -3\\
-3 & 3
\end{bmatrix}
$$
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Multiplication of matrices
The product of two matrices A and B can be defined if the number of rows of B is equal to the number of columns of A i.e. if A be an m × n matrix and B be an n × p matrix then the product of matrices A and B is another matrix of order m × p.
Example: Multiplication of Two Matrices
Let
$$
A=
\begin{bmatrix}
2 & 3\\
1 & 4
\end{bmatrix},
\qquad
B=
\begin{bmatrix}
5 & 2\\
6 & 1
\end{bmatrix}
$$
Then,
$$
AB=
\begin{bmatrix}
2 & 3\\
1 & 4
\end{bmatrix}
\begin{bmatrix}
5 & 2\\
6 & 1
\end{bmatrix}
$$
Compute each entry:
$$
AB=
\begin{bmatrix}
(2\times5+3\times6) & (2\times2+3\times1)\\
(1\times5+4\times6) & (1\times2+4\times1)
\end{bmatrix}
$$
$$
=
\begin{bmatrix}
28 & 7\\
29 & 6
\end{bmatrix}
$$
Hence,
$$
AB=
\begin{bmatrix}
28 & 7\\
29 & 6
\end{bmatrix}
$$
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Transpose of a Matrix
If A = [aij] be an m × n matrix, then the matrix obtained by interchanging the rows and columns of A is called the transpose of A. Transpose of the matrix A is denoted by A’ or AT.
Example: Transpose of a Matrix
Let
$$
A=
\begin{bmatrix}
2 & 5 & 7\\
1 & 4 & 6
\end{bmatrix}
$$
The transpose of a matrix is obtained by interchanging its rows and columns.
$$
A^T=
\begin{bmatrix}
2 & 1\\
5 & 4\\
7 & 6
\end{bmatrix}
$$
Hence,
$$
\left(
\begin{bmatrix}
2 & 5 & 7\\
1 & 4 & 6
\end{bmatrix}
\right)^T
=
\begin{bmatrix}
2 & 1\\
5 & 4\\
7 & 6
\end{bmatrix}
$$
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Properties of transpose of the Matrices
\[
(A^{T})^{T} = A
\]
\[
(kA)^{T} = kA^{T}
\]
\[
(A + B)^{T} = A^{T} + B^{T}
\]
\[
(AB)^{T} = B^{T}A^{T}
\]
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Symmetric Matrix
A square matrix M is said to be symmetric if AT = A
Note: There will be symmetry about the principal diagonal in Symmetric Matrix.
Example: Symmetric Matrix
A matrix is called Symmetric if
$$
A^T=A
$$
Consider the matrix
$$
A=
\begin{bmatrix}
2 & 3 & 5\\
3 & 4 & 6\\
5 & 6 & 7
\end{bmatrix}
$$
Its transpose is
$$
A^T=
\begin{bmatrix}
2 & 3 & 5\\
3 & 4 & 6\\
5 & 6 & 7
\end{bmatrix}
$$
Since
$$
A^T=A,
$$
Therefore, the matrix \(A\) is a Symmetric Matrix.
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Skew symmetric Matrix
A square matrix M is said to be skew symmetric if AT = -A
Note: All the principal diagonal element of a skew symmetric Matrix are zero.
Example: Skew-Symmetric Matrix
A matrix is called Skew-Symmetric if
$$
A^T=-A
$$
Consider the matrix
$$
A=
\begin{bmatrix}
0 & 2 & -3\\
-2 & 0 & 5\\
3 & -5 & 0
\end{bmatrix}
$$
Its transpose is
$$
A^T=
\begin{bmatrix}
0 & -2 & 3\\
2 & 0 & -5\\
-3 & 5 & 0
\end{bmatrix}
$$
Also,
$$
-A=
\begin{bmatrix}
0 & -2 & 3\\
2 & 0 & -5\\
-3 & 5 & 0
\end{bmatrix}
$$
Since
$$
A^T=-A,
$$
Therefore, the matrix \(A\) is a Skew-Symmetric Matrix.