Chapter 3 – Matrices and Determinants
1. Matrix
A matrix is a rectangular arrangement of numbers in rows and columns.
2. Order of a Matrix
If a matrix has $m$ rows and $n$ columns, its order is $m \times n$.
3. Row Matrix
A matrix having only one row is called a row matrix.
4. Column Matrix
A matrix having only one column is called a column matrix.
5. Square Matrix
If number of rows = number of columns, matrix is a square matrix.
6. Zero Matrix
A matrix in which all elements are zero is called zero matrix.
7. Diagonal Matrix
A square matrix in which all non-diagonal elements are zero.
8. Identity Matrix
A diagonal matrix in which all diagonal elements are 1.
9. Equality of Matrices
Two matrices are equal if they have same order and corresponding elements are equal.
10. Addition of Matrices
Addition is possible only when matrices have same order.
11. Scalar Multiplication
Each element of matrix is multiplied by a scalar.
12. Matrix Multiplication
If $A$ is $m\times n$ and $B$ is $n\times p$, then $AB$ exists and is $m\times p$.
13. Non-Commutativity
Matrix multiplication is not commutative i.e. $AB \ne BA$.
14. Transpose of Matrix
If $A=[a_{ij}]$, then $A^T=[a_{ji}]$
15. Symmetric Matrix
A square matrix is symmetric if $A=A^T$.
16. Skew-Symmetric Matrix
A square matrix is skew-symmetric if $A^T=-A$.
17. Determinant
Determinant is a real number associated with a square matrix.
18. Determinant of $2\times2$
$\begin{vmatrix}a & b \\ c & d\end{vmatrix} = ad-bc$
19. Properties of Determinants
- Interchanging rows changes sign
- Two equal rows → determinant is zero
- Row multiplied by $k$ → determinant multiplied by $k$
20. Inverse of a Matrix
$A^{-1} = \dfrac{\text{Adj}(A)}{|A|}$, if $|A|\ne0$
Important Questions
Q1. Find determinant of $\begin{vmatrix}1 & 2 \\ 3 & 4\end{vmatrix}$
$= (1)(4)-(2)(3) = -2$
Q2. If $|A|=5$, find $|A^{-1}|$
$= \dfrac{1}{5}$
Q3. Can a matrix have inverse if determinant is zero?
No
Q4. If $A$ is skew-symmetric, find $a_{ii}$
$a_{ii}=0$
Q5. Is matrix multiplication commutative?
No
Q6. Find order of product if $A$ is $2\times3$ and $B$ is $3\times4$
$2\times4$
Q7. What is determinant of identity matrix?
1
Q8. If two rows are same, determinant value?
0
Q9. Define symmetric matrix
$A=A^T$
Q10. Find transpose of $\begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix}$
$\begin{bmatrix}1 & 3 \\ 2 & 4\end{bmatrix}$
Q11. What is zero matrix?
Matrix with all elements zero
Q12. Determinant of $\begin{vmatrix}a & b \\ ka & kb\end{vmatrix}$
0
Q13. If $A$ is identity matrix, find $A^{-1}$
$A$
Q14. Can $AB=0$ but $A\ne0,B\ne0$?
Yes
Q15. Formula for inverse of matrix
$\dfrac{\text{Adj}(A)}{|A|}$