If \([A]_{3 \times 2} [B]_{x \times y} = [C]_{3 \times 1}\), then:
\( x = 2, y = 1 \)
Understanding Matrix Multiplication Dimensions
Matrix multiplication has specific rules regarding the dimensions of the matrices involved. For two matrices to be multiplied, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix.
Analyzing the Given Matrix Equation
We are given the matrix equation \([A]_{3 \times 2} [B]_{x \times y} = [C]_{3 \times 1}\). This means matrix A is multiplied by matrix B to produce matrix C.
- Matrix A has dimensions \(3 \times 2\) (3 rows and 2 columns).
- Matrix B has dimensions \(x \times y\) (x rows and y columns).
- Matrix C has dimensions \(3 \times 1\) (3 rows and 1 column).
Applying Matrix Multiplication Rules to Find x and y
Let's apply the rules of matrix multiplication to determine the values of \(x\) and \(y\).
Condition for Multiplication
For the product \(A \times B\) to be defined, the number of columns in matrix A must equal the number of rows in matrix B.
Number of columns in A = 2
Number of rows in B = \(x\)
Therefore, according to the rule:
\(2 = x\)
So, the value of \(x\) must be 2.
Dimensions of the Resulting Matrix
The resulting matrix C has dimensions determined by the number of rows in A and the number of columns in B.
Number of rows in C = Number of rows in A = 3
Number of columns in C = Number of columns in B = \(y\)
From the given dimensions of C (\(3 \times 1\)), we know:
Number of rows in C = 3 (This matches the number of rows in A, as expected)
Number of columns in C = 1
Therefore, according to the rule:
\(1 = y\)
So, the value of \(y\) must be 1.
Determining the Dimensions of Matrix B
Based on our analysis, the dimensions of matrix B are \(x \times y\), where \(x=2\) and \(y=1\). Thus, matrix B has dimensions \(2 \times 1\).
Comparing with the Options
Let's compare our result with the given options:
- Option 1: \(x = 1, y = 3\) (Incorrect, does not match \(x=2, y=1\))
- Option 2: \(x = 2, y = 1\) (Correct, matches our derived values)
- Option 3: \(x = 3, y = 3\) (Incorrect, does not match \(x=2, y=1\))
- Option 4: \(x = 3, y = 1\) (Incorrect, does not match \(x=2, y=1\))
The dimensions \(x=2\) and \(y=1\) are required for the matrix multiplication \([A]_{3 \times 2} [B]_{x \times y}\) to be defined and to result in a matrix \([C]_{3 \times 1}\).
| Matrix | Given Dimensions | Rows | Columns |
|---|---|---|---|
| A | \(3 \times 2\) | 3 | 2 |
| B | \(x \times y\) | \(x\) | \(y\) |
| C | \(3 \times 1\) | 3 | 1 |
| Matrix Multiplication Rule | Application | Result |
|---|---|---|
| Columns of first matrix = Rows of second matrix | Columns of A = Rows of B \(2 = x\) | \(x=2\) |
| Resulting matrix rows = Rows of first matrix | Rows of C = Rows of A \(3 = 3\) | Consistent |
| Resulting matrix columns = Columns of second matrix | Columns of C = Columns of B \(1 = y\) | \(y=1\) |
Revision Table: Matrix Dimensions for Multiplication
| Operation | Matrix 1 (m x n) | Matrix 2 (p x q) | Condition for Multiplication | Resulting Matrix Dimensions |
|---|---|---|---|---|
| Multiplication (\(Matrix_1 \times Matrix_2\)) | m x n | p x q | \(n = p\) | m x q |
Additional Information: Matrix Product Properties
Understanding matrix dimensions is crucial for performing matrix operations. Here are a few key points about matrix multiplication:
- Matrix multiplication is not commutative in general, meaning \(A \times B\) is not always equal to \(B \times A\). Even if both products are defined, their results may be different.
- The product \(A \times B\) is defined only if the number of columns in \(A\) equals the number of rows in \(B\).
- The dimensions of the resulting matrix \(C = A \times B\) are determined by the number of rows of the first matrix (\(A\)) and the number of columns of the second matrix (\(B\)).
- Matrix multiplication is associative: \((A \times B) \times C = A \times (B \times C)\), provided the dimensions allow the operations.
- Matrix multiplication is distributive over matrix addition: \(A \times (B + C) = A \times B + A \times C\) and \((A + B) \times C = A \times C + B \times C\), provided the dimensions allow the operations.
Important Questions from Matrices
Three defective bulbs are mixed with 8 good ones. If three bulbs are drawn one by one with replacement, the probabilities of getting exactly 1 defective, more than 2 defective, no defective and more than 1 defective respectively are :
If A, B, and C are three singular matrices given by:
\[ A = \begin{bmatrix} 1 & 4 \\ 3 & 2a \end{bmatrix} \]
\[ B = \begin{bmatrix} 3b & 5 \\ a & 2 \end{bmatrix} \]
and
\[ C = \begin{bmatrix} a + b + c & c + 1 \\ a + c & c \end{bmatrix} \]
then the value of abc is:
If \( P = \begin{bmatrix} -1 \\ 2 \\ 1 \end{bmatrix} \) and \( Q = \begin{bmatrix} 2 & -4 & 1 \end{bmatrix} \) are two matrices, then \( (PQ)' \) will be:
The matrix
\[ \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \]
is a:
- \( \text{Scalar matrix} \)
- \( \text{Diagonal matrix} \)
- \( \text{Skew-symmetric matrix} \)
- \( \text{Symmetric matrix} \)
Choose the correct answer from the options given below:
If A is a square matrix and I is an identity matrix such that \(A^2 = A\), then \(A(I - 2A)^3 + 2A^3\) is equal to :
