Determinants and Matrices

Matrices are grids of numbers organized in rows and columns. They're used to solve equations and describe transformations in math.

Determinants are special numbers calculated from square matrices. They help find areas, volumes, and solve equations in geometry and algebra.

1.0What Are Matrices and Determinants?

A matrix is a structured arrangement of numbers or functions, organized in rows and columns. Each individual number or function within the matrix is referred to as an element or entry.

A determinant is a scalar value derived from a square matrix. It provides essential information about the matrix, such as whether it is invertible (non-singular) or not (singular). Determinants are used in solving systems of linear equations, finding areas and volumes, and in various calculations in geometry and algebra. They play a crucial role in determining the properties of linear transformations represented by matrices.

2.0Difference Between Matrices and Determinants

3.0Properties of Matrices and Determinants

4.0Matrices and Determinants Solved Examples 

Example 1: Find the value of x, y, z and w which satisfy the matrix equation

$\left[\begin{array}{ll}x+3& 2y+x\\ z-1& 4w-8\end{array}\right]=\left[\begin{array}{cc}-x-1& 0\\ 3& 2w\end{array}\right]$

Solution:

As the given matrices are equal so their corresponding elements are equal.

$\Rightarrow$ x+3=-x-1

$\Rightarrow$ 2 x=-4

$\therefore$ x=-2 ...(i)

$\Rightarrow$ 2 y+x=0

$\Rightarrow$ 2 y-2=0 [from (i)]

$\Rightarrow$ y=1 ...(ii)

$\Rightarrow$ z-1=3

$\Rightarrow$ z=4 ...(iii)

$\Rightarrow$ 4 w-8=2 w

$\Rightarrow$ 2 w=8

$\therefore$ w=4 ...(iv)

Example 2: If A=$\left[\begin{array}{ll}1& 3\\ 3& 2\\ 2& 5\end{array}\right]\&B=\left[\begin{array}{cc}-1& -2\\ 0& 5\\ 3& 1\end{array}\right]\text{ and }A+B-D=\mathrm{O}$

(zero matrix), then D matrix will be-

(A) $\left[\begin{array}{ll}0& 2\\ 3& 7\\ 6& 5\end{array}\right]$

(B) $\left[\begin{array}{ll}0& 2\\ 3& 7\\ 5& 6\end{array}\right]$

(C) $\left[\begin{array}{ll}0& 1\\ 3& 7\\ 5& 6\end{array}\right]$

(D) $\left[\begin{array}{cc}0& -2\\ -3& -7\\ -5& -6\end{array}\right]$

Ans. (C)

Solution:

Let D=

$\left[\begin{array}{ll}a& b\\ c& d\\ e& f\end{array}\right]\therefore A+B-D=\left[\begin{array}{ll}1& 3\\ 3& 2\\ 2& 5\end{array}\right]+\left[\begin{array}{cc}-1& -2\\ 0& 5\\ 3& 1\end{array}\right]-\left[\begin{array}{ll}a& b\\ c& d\\ e& f\end{array}\right]$

$\Rightarrow \left[\begin{array}{ll}1-1-a& 3-2-b\\ 3+0-c& 2+5-d\\ 2+3-e& 5+1-f\end{array}\right]=\left[\begin{array}{ll}0& 0\\ 0& 0\\ 0& 0\end{array}\right]$

$\Rightarrow -a=0\Rightarrow a=0,1-b=0\Rightarrow b=1\text{, }\Rightarrow 3-c=0\Rightarrow c=3,7-d=0\Rightarrow d=7\text{, }\Rightarrow 5-e=0\Rightarrow e=5,6-f=0\Rightarrow f=6$

$\therefore D=\left[\begin{array}{ll}0& 1\\ 3& 7\\ 5& 6\end{array}\right]$

Example 3: If $[1\times 2]\left[\begin{array}{lll}2& 3& 1\\ 0& 4& 2\\ 0& 3& 2\end{array}\right]\left[\begin{array}{r}x\\ 1\\ -1\end{array}\right]$=O, then the value of is

(A) –1 (B) 0 (C) 1 (D) 2

Ans. (A)

Solution:

The LHS of the equation

=$[24x+92x+5]\left[\begin{array}{r}x\\ 1\\ -1\end{array}\right]=[2x+4x+9-2x-5]=4x+4$

=$\left[\begin{array}{lll}2& 4x+9& 2x+5\end{array}\right]\left[\begin{array}{r}x\\ 1\\ -1\end{array}\right]=[2x+4x+9-2x-5]=[4x+4]$

Thus 4 x+4=0 ; x=-1

Example 4: If A, B are two matrices such that A+B= $\left[\begin{array}{ll}1& 2\\ 2& 4\end{array}\right],A-B=\left[\begin{array}{cc}3& 2\\ -2& 0\end{array}\right]$then find AB.

Solution:

Given A+B=$\left[\begin{array}{ll}1& 2\\ 2& 4\end{array}\right]$         ...(i)     &

$\Rightarrow A-B=\left[\begin{array}{cc}3& 2\\ -2& 0\end{array}\right]$ ...(ii)

Adding (i) & (ii)

$\Rightarrow 2A=\left[\begin{array}{ll}4& 4\\ 0& 4\end{array}\right]\Rightarrow A=\left[\begin{array}{ll}2& 2\\ 0& 2\end{array}\right]$

Subtracting (ii) from (i)

$\Rightarrow 2B=\left[\begin{array}{cc}-2& 0\\ 4& 4\end{array}\right]\Rightarrow B=\left[\begin{array}{cc}-1& 0\\ 2& 2\end{array}\right]$

Now $AB=\left[\begin{array}{ll}2& 2\\ 0& 2\end{array}\right]\left[\begin{array}{cc}-1& 0\\ 2& 2\end{array}\right]=\left[\begin{array}{ll}2& 4\\ 4& 4\end{array}\right]$

Example 5: The value of $\left|\begin{array}{ccc}1& 2& 3\\ -4& 3& 6\\ 2& -7& 9\end{array}\right|$ is  

(A) 213 (B) –231 (C) 231 (D) 39 

Ans. (C)

Solution: 

$\left|\begin{array}{ccc}1& 2& 3\\ -4& 3& 6\\ 2& -7& 9\end{array}\right|=1\left|\begin{array}{cc}3& 6\\ -7& 9\end{array}\right|-2\left|\begin{array}{cc}-4& 6\\ 2& 9\end{array}\right|+3\left|\begin{array}{cc}-4& 3\\ 2& -7\end{array}\right|$

$=(27+42)-2(-36-12)+3(28-6)=231$

Example 6: If $\left|\begin{array}{ccc}x+3& 1& -2\\ 3& -2& 1\\ -x& -3& 3\end{array}\right|$=0, find $\chi$.

Solution:

$\left|\begin{array}{ccc}x+3& 1& -2\\ 3& -2& 1\\ -x& -3& 3\end{array}\right|=0$

$\Rightarrow (x+3)[-6-(-3)]-1[9+x]-2[-9-2x]$

$\Rightarrow (x+3)(-6+3)-(9+x)-2(-9-2x)=0$

$\Rightarrow -3(x+3)-9-x+18+4x=0$

$\Rightarrow -3x-9-9-x+18+4x=0$

$\Rightarrow -4x+4x-18+18=0$

$\therefore x\in R$

Example 7: If in the determinant $\left|\begin{array}{lll}x& 3& 3\\ 3& 3& x\\ 2& 3& 3\end{array}\right|,C_{11}=C_{22}$, where $C_{ij}$ is cofactor of element $a_{ij}$ then x= 

(A) 2 (B) –2 (C)  $\frac{5}{2}$ (D) $-\frac{9}{8}$

Ans. (C)

Solution:

$C_{11}=C_{22}$

$\models (-1{)}^{1+1}\left|\begin{array}{ll}3& x\\ 3& 3\end{array}\right|=(-1{)}^{2+2}\left|\begin{array}{ll}x& 3\\ 2& 3\end{array}\right|$

$\Rightarrow 9-3x=3x-6$

$\Rightarrow -6x=-15$

$\Rightarrow x=\frac{5}{2}$

Example 8: If $\left|\begin{array}{lll}{3}^2+k& 4^2& 3^2+3+k\\ 4^2+k& 5^2& 4^2+4+k\\ 5^2+k& 6^2& 5^2+5+k\end{array}\right|$=0,$\text{ then the value of }k$ is?

(A) 2 (B) 1 (C) –1 (D) 0

Ans. (B)

Solution: 

Applying $\left(C_3\to C_3-C_1\right)D=\left|\begin{array}{ccc}{3}^2+k& 4^2& 3\\ 4^2+k& 5^2& 4\\ 5^2+k& 6^2& 5\end{array}\right|$=0

$\Rightarrow \left|\begin{array}{ccc}9+k& 16& 3\\ 7& 9& 1\\ 9& 11& 1\end{array}\right|=0\left(R_3\to R_3-R_2;R_2\to R_2-R_1\right)$

$\Rightarrow k-1=0$

$\Rightarrow k=1$

5.0Matrices and Determinants Practice problems

1. If $\left[\begin{array}{ccc}2x+y& 2& x-2y\\ a-b& 2a+b& -3\end{array}\right]=\left[\begin{array}{ccc}3& 2& 4\\ 4& -1& -3\end{array}\right]$, then find the value of x+y+a+b

(A) 1 (B) –1 (C) –3 (D) 5

2. Find the value of :

$2\left[\begin{array}{ccc}3& 1& -2\\ -1& -3& 4\end{array}\right]+x\left[\begin{array}{ccc}1& 0& -1\\ 3& 4& 5\end{array}\right]=\left[\begin{array}{ccc}4& 2& -2\\ -8& -14& -2\end{array}\right]$

(A) –2 (B) 2 (C) 3 (D) –3

3. Let A=$\left[\begin{array}{lll}{\cos }^{-1}x& {\cos }^{-1}y& {\cos }^{-1}z\\ {\cos }^{-1}y& {\cos }^{-1}z& {\cos }^{-1}x\\ {\cos }^{-1}z& {\cos }^{-1}x& {\cos }^{-1}y\end{array}\right]$such that |A|=0, then maximum value of x+y+z is 

(A) 3 (B) 0 (C) 1 (D) 2

4. If α, β & γ are the roots of the equation $x^3+px+q=0$, then the value of the determinant $\left|\begin{array}{lll}\alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{array}\right|$=

(A) p (B) q (C) p^2 \_2 q (D) none

5. If $a,b\&c$ are non-zero real numbers^, then D=$\left|\begin{array}{lll}{b}^2{c}^2& bc& b+c\\ c^2{a}^2& ca& c+a\\ a^2{b}^2& ab& a+b\end{array}\right|$=

(A) abc (B) a²b²c² (C) bc + ca +ab   (D) zero

6.0Sample Questions on Determinants and Matrices

Q. How do you add two matrices?

Ans: To add two matrices, they must have the same dimensions. You add corresponding elements from each matrix. For example, given matrices A and B:

A=$\left[\begin{array}{ll}1& 2\\ 3& 4\end{array}\right],B=\left[\begin{array}{ll}5& 6\\ 7& 8\end{array}\right]$ 

The sum A + B is:

A+B=$\left[\begin{array}{ll}1+5& 2+6\\ 3+7& 4+8\end{array}\right]=\left[\begin{array}{cc}6& 8\\ 10& 12\end{array}\right]$

Q. How is the determinant of a matrix calculated?

Ans: For a 2 × 2 matrix $\left(\begin{array}{ll}a& b\\ c& d\end{array}\right)$, the determinant is calculated as ad – bc. For larger matrices, determinants can be calculated using cofactor expansion, row reduction, or other methods. For example, for a 3 × 3 matrix $\left(\begin{array}{lll}a& b& c\\ d& e& f\\ g& h& i\end{array}\right)$, the determinant is a(ei – fh) – b(di – fg) + c(dh – eg).

Q. What is the trace of a matrix?

Ans: The trace of a square matrix is the sum of its diagonal elements. For example, for the matrix \left(\begin{array}{ll}1 & 2 \\3 & 4\end{array}\right), the trace is 1 + 4 = 5. The trace has various applications, particularly in the study of eigenvalues and matrix invariants.