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Q7:
Find the value of $a, b, c$ and $d$ from the equation: $\begin{bmatrix} a-b & 2a+c \\ 2a-b & 3c+d \end{bmatrix} = \begin{bmatrix} -1 & 5 \\ 0 & 13 \end{bmatrix}$
Solution :
Step 1: Establishing the System of Linear Equations
By the definition of equality of matrices, two matrices are equal if and only if their corresponding elements are identical [Per Definition of Matrix Equality]. Given the equation:
$\begin{bmatrix} a-b & 2a+c \\ 2a-b & 3c+d \end{bmatrix} = \begin{bmatrix} -1 & 5 \\ 0 & 13 \end{bmatrix}$
We equate the corresponding entries to form a system of four linear equations:
- Eq (i): $a - b = -1$
- Eq (ii): $2a + c = 5$
- Eq (iii): $2a - b = 0$
- Eq (iv): $3c + d = 13$
Step 2: Solving for Variables $a$ and $b$
Utilizing Eq (i) and Eq (iii), we perform elimination [By Cramer’s Rule/Elimination Method]:
From Eq (iii), we isolate $b$: $b = 2a$.
Substitute $b = 2a$ into Eq (i):
$a - (2a) = -1$
$-a = -1 \implies a = 1$
Now, substitute $a = 1$ back into the expression for $b$:
$b = 2(1) = 2$
Step 3: Solving for Variables $c$ and $d$
Substitute $a = 1$ into Eq (ii) to determine $c$:
$2(1) + c = 5$
$2 + c = 5 \implies c = 3$
Now, substitute $c = 3$ into Eq (iv) to determine $d$:
$3(3) + d = 13$
$9 + d = 13$
$d = 13 - 9 = 4$
Visual Representation of Matrix Mapping
Step 4: Summary Table of Results
| Variable | Value |
|---|---|
| $a$ | $1$ |
| $b$ | $2$ |
| $c$ | $3$ |
| $d$ | $4$ |
Final Solution: The values are $a=1, b=2, c=3, d=4$.
More Questions from Class 12 Mathematics Matrices EXERCISE 3.1
- Q1(i): In the matrix $A = \begin{bmatrix} 2 & 5 & 19 & -7 \\ 35 & -2 & \frac{5}{2} & 12 \\ \sqrt{3} & 1 & -5 & 17 \end{bmatrix}$, write: (i) The order of the matrix
- Q1(ii): In the matrix $A = \begin{bmatrix} 2 & 5 & 19 & -7 \\ 35 & -2 & \frac{5}{2} & 12 \\ \sqrt{3} & 1 & -5 & 17 \end{bmatrix}$, write: (ii) The number of elements
- Q1(iii): In the matrix $A = \begin{bmatrix} 2 & 5 & 19 & -7 \\ 35 & -2 & \frac{5}{2} & 12 \\ \sqrt{3} & 1 & -5 & 17 \end{bmatrix}$, write: (iii) Write the elements $a_{13}, a_{21}, a_{33}, a_{24}, a_{23}$.
- Q2: If a matrix has 24 elements, what are the possible orders it can have? What, if it has 13 elements?
- Q3: If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?
- Q4(i): Construct a $2 \times 2$ matrix, $A = [a_{ij}]$, whose elements are given by: (i) $a_{ij} = \frac{(i+j)^2}{2}$
- Q4(ii): Construct a $2 \times 2$ matrix, $A = [a_{ij}]$, whose elements are given by: (ii) $a_{ij} = \frac{i}{j}$
- Q4(iii): Construct a $2 \times 2$ matrix, $A = [a_{ij}]$, whose elements are given by: (iii) $a_{ij} = \frac{(i+2j)^2}{2}$
- Q5(i): Construct a $3 \times 4$ matrix, whose elements are given by: (i) $a_{ij} = \frac{1}{2}|-3i + j|$
- Q5(ii): Construct a $3 \times 4$ matrix, whose elements are given by: (ii) $a_{ij} = 2i-j$
- Q6(i): Find the values of $x, y$ and $z$ from the following equations: (i) $\begin{bmatrix} 4 & 3 \\ x & 5 \end{bmatrix} = \begin{bmatrix} y & z \\ 1 & 5 \end{bmatrix}$
- Q6(ii): Find the values of $x, y$ and $z$ from the following equations: (ii) $\begin{bmatrix} x+y & 2 \\ 5+z & xy \end{bmatrix} = \begin{bmatrix} 6 & 2 \\ 5 & 8 \end{bmatrix}$
- Q6(iii): Find the values of $x, y$ and $z$ from the following equations: (iii) $\begin{bmatrix} x+y+z \\ x+z \\ y+z \end{bmatrix} = \begin{bmatrix} 9 \\ 5 \\ 7 \end{bmatrix}$
- Q8: $A = [a_{ij}]_{m \times n}$ is a square matrix, if
- Q9: Which of the given values of $x$ and $y$ make the following pair of matrices equal $\begin{bmatrix} 3x+7 & 5 \\ y+1 & 2-3x \end{bmatrix}$, $\begin{bmatrix} 0 & y-2 \\ 8 & 4 \end{bmatrix}$
- Q10: The number of all possible matrices of order $3 \times 3$ with each entry 0 or 1 is:
CBSE Solutions for Class 12 Mathematics Matrices
Chapters in CBSE - Class 12 Mathematics
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