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Q4:
For what values of $x$ : $\begin{bmatrix} 1 & 2 & 1 \end{bmatrix} \begin{bmatrix} 1 & 2 & 0 \\ 2 & 0 & 1 \\ 1 & 0 & 2 \end{bmatrix} \begin{bmatrix} 0 \\ 2 \\ x \end{bmatrix} = O$?
Solution :
Step 1: Establishing the Matrix Multiplication Sequence
Given the matrix equation: $\begin{bmatrix} 1 & 2 & 1 \end{bmatrix} \begin{bmatrix} 1 & 2 & 0 \\ 2 & 0 & 1 \\ 1 & 0 & 2 \end{bmatrix} \begin{bmatrix} 0 \\ 2 \\ x \end{bmatrix} = [0]$
We apply the associative property of matrix multiplication, $(AB)C = A(BC)$, to simplify the calculation [Per Associative Property of Matrices]. Let $A = \begin{bmatrix} 1 & 2 & 1 \end{bmatrix}$, $B = \begin{bmatrix} 1 & 2 & 0 \\ 2 & 0 & 1 \\ 1 & 0 & 2 \end{bmatrix}$, and $C = \begin{bmatrix} 0 \\ 2 \\ x \end{bmatrix}$.
Step 2: Multiplying the Second and Third Matrices
Let $D = B \times C$. $D = \begin{bmatrix} 1 & 2 & 0 \\ 2 & 0 & 1 \\ 1 & 0 & 2 \end{bmatrix} \begin{bmatrix} 0 \\ 2 \\ x \end{bmatrix}$
Perform the row-by-column multiplication: $D = \begin{bmatrix} (1 \cdot 0) + (2 \cdot 2) + (0 \cdot x) \\ (2 \cdot 0) + (0 \cdot 2) + (1 \cdot x) \\ (1 \cdot 0) + (0 \cdot 2) + (2 \cdot x) \end{bmatrix}$
$D = \begin{bmatrix} 0 + 4 + 0 \\ 0 + 0 + x \\ 0 + 0 + 2x \end{bmatrix} = \begin{bmatrix} 4 \\ x \\ 2x \end{bmatrix}$
Step 3: Multiplying by the First Matrix
Now, compute $A \times D = [0]$: $\begin{bmatrix} 1 & 2 & 1 \end{bmatrix} \begin{bmatrix} 4 \\ x \\ 2x \end{bmatrix} = [0]$
Expand the dot product: $(1 \cdot 4) + (2 \cdot x) + (1 \cdot 2x) = 0$
Step 4: Solving the Linear Algebraic Equation
Simplify the resulting scalar equation: $4 + 2x + 2x = 0$
$4 + 4x = 0$
Isolate the variable $x$: $4x = -4$
$x = \frac{-4}{4}$
$x = -1$
Visual Representation of the Operation
Final Solution: $x = -1$
More Questions from Class 12 Mathematics Matrices Miscellaneous Exercise on Chapter 3
- Q1: If A and B are symmetric matrices, prove that AB β BA is a skew symmetric matrix.
- Q2: Show that the matrix Bβ²AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.
- Q3: Find the values of $x, y, z$ if the matrix $A = \begin{bmatrix} 0 & 2y & z \\ x & y & -z \\ x & -y & z \end{bmatrix}$ satisfy the equation $Aβ²A = I$.
- Q5: If $A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}$, show that $A^2 β 5A + 7I = 0$.
- Q6: Find $x$, if $\begin{bmatrix} x & -5 & -1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{bmatrix} \begin{bmatrix} x \\ 4 \\ 1 \end{bmatrix} = O$
- Q7(a): A manufacturer produces three products $x, y, z$ which he sells in two markets. Annual sales are indicated below: (a) If unit sale prices of $x, y$ and $z$ are ` 2.50, ` 1.50 and ` 1.00, respectively, find the total revenue in each market with the help of matrix algebra.
- Q7(b): A manufacturer produces three products $x, y, z$ which he sells in two markets. Annual sales are indicated below: (b) If the unit costs of the above three commodities are ` 2.00, ` 1.00 and 50 paise respectively. Find the gross profit.
- Q8: Find the matrix $X$ so that $X \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} = \begin{bmatrix} -7 & -8 & -9 \\ 2 & 4 & 6 \end{bmatrix}$
- Q9: If $A = \begin{bmatrix} \alpha & \beta \\ \gamma & -\alpha \end{bmatrix}$ is such that $A^2 = I$, then
- Q10: If the matrix A is both symmetric and skew symmetric, then
- Q11: If A is square matrix such that $A^2 = A$, then $(I + A)^3 β 7 A$ is equal to
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