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Q6(i):
Find the values of $x, y$ and $z$ from the following equations: (i) $\begin{bmatrix} 4 & 3 \\ x & 5 \end{bmatrix} = \begin{bmatrix} y & z \\ 1 & 5 \end{bmatrix}$
Solution :
Step 1: Establishing the Equality of Matrices
Two matrices $A = [a_{ij}]$ and $B = [b_{ij}]$ are said to be equal if and only if:
- They are of the same order.
- Their corresponding elements are equal for all possible values of $i$ and $j$ ($a_{ij} = b_{ij}$).
[Definition of Equality of Matrices]
Step 2: Verification of Matrix Order
The given matrices are:
$A = \begin{bmatrix} 4 & 3 \\ x & 5 \end{bmatrix}$ (Order: $2 \times 2$)
$B = \begin{bmatrix} y & z \\ 1 & 5 \end{bmatrix}$ (Order: $2 \times 2$)
Since both matrices have an order of $2 \times 2$, the equality is well-defined. [Per the condition of equality for matrices of same dimensions].
Step 3: Equating Corresponding Elements
By equating the elements at each position $(i, j)$ from matrix $A$ to matrix $B$:
| Position $(i, j)$ | Equation |
|---|---|
| Row 1, Column 1 | $4 = y$ |
| Row 1, Column 2 | $3 = z$ |
| Row 2, Column 1 | $x = 1$ |
| Row 2, Column 2 | $5 = 5$ (Identity holds) |
Visual Representation of Matrix Equality
Step 4: Final Derivation
From the equations derived in Step 3, we directly extract the values:
- $x = 1$
- $y = 4$
- $z = 3$
Final Solution: x = 1, y = 4, z = 3
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(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) $\begin{bmatrix} x+y+z \\ x+z \\ y+z \end{bmatrix} = \begin{bmatrix} 9 \\ 5 \\ 7 \end{bmatrix}$
- Q7: Find the value of $a, b, c$ and $d$ from the equation: $\begin{bmatrix} a-b & 2a+c \\ 2a-b & 3c+d \end{bmatrix} = \begin{bmatrix} -1 & 5 \\ 0 & 13 \end{bmatrix}$
- 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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