If \( A = \begin{bmatrix} x & -y \\ y & x \end{bmatrix} \) and \( A + A' = 2I \), then \( x \) is equal to:
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Solving the Matrix Equation \( A + A' = 2I \)
We are given a matrix \( A \) and the equation \( A + A' = 2I \), where \( A' \) is the transpose of \( A \) and \( I \) is the identity matrix. We need to find the value of \( x \) using this information.
Understanding the Given Matrix and its Transpose
The given matrix is:
\[ A = \begin{bmatrix} x & -y \\ y & x \end{bmatrix} \]The transpose of a matrix \( A \), denoted by \( A' \) or \( A^T \), is obtained by interchanging its rows and columns. For the given matrix \( A \), the first row becomes the first column and the second row becomes the second column.
So, the transpose \( A' \) is:
\[ A' = \begin{bmatrix} x & y \\ -y & x \end{bmatrix} \]Understanding the Identity Matrix
The identity matrix \( I \) is a square matrix with ones on the main diagonal and zeros elsewhere. Since matrix \( A \) is a 2x2 matrix, the identity matrix \( I \) relevant here is the 2x2 identity matrix:
\[ I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \]The equation involves \( 2I \), which means we multiply each element of the identity matrix by 2:
\[ 2I = 2 \times \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 2 \times 1 & 2 \times 0 \\ 2 \times 0 & 2 \times 1 \end{bmatrix} = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix} \]Setting up the Matrix Equation \( A + A' = 2I \)
Now we substitute the matrices \( A \), \( A' \), and \( 2I \) into the given equation:
\[ \begin{bmatrix} x & -y \\ y & x \end{bmatrix} + \begin{bmatrix} x & y \\ -y & x \end{bmatrix} = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix} \]Performing Matrix Addition
To add matrices, we add the corresponding elements:
\[ \begin{bmatrix} x + x & -y + y \\ y + (-y) & x + x \end{bmatrix} = \begin{bmatrix} 2x & 0 \\ 0 & 2x \end{bmatrix} \]Equating the Matrices and Solving for x
Now we have the resulting sum matrix equal to \( 2I \):
\[ \begin{bmatrix} 2x & 0 \\ 0 & 2x \end{bmatrix} = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix} \]For two matrices to be equal, their corresponding elements must be equal. Comparing the elements, we get the following equations:
- Top-left element: \( 2x = 2 \)
- Top-right element: \( 0 = 0 \)
- Bottom-left element: \( 0 = 0 \)
- Bottom-right element: \( 2x = 2 \)
From the equations involving \( x \), we have:
\[ 2x = 2 \]To find \( x \), we divide both sides by 2:
\[ x = \frac{2}{2} \] \[ x = 1 \]The value of \( y \) is not needed to solve for \( x \), and the terms involving \( y \) correctly cancel out in the matrix addition, resulting in zero elements where expected in the identity matrix.
Final Answer
The value of \( x \) that satisfies the equation \( A + A' = 2I \) is 1.
Revision Table: Key Matrix Concepts
| Concept | Description | Example (2x2) |
|---|---|---|
| Matrix Addition | Adding corresponding elements of two matrices of the same size. | \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} + \begin{bmatrix} e & f \\ g & h \end{bmatrix} = \begin{bmatrix} a+e & b+f \\ c+g & d+h \end{bmatrix} \) |
| Matrix Transpose (\( A' \) or \( A^T \)) | Swapping rows and columns of a matrix. | If \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \), then \( A' = \begin{bmatrix} a & c \\ b & d \end{bmatrix} \) |
| Scalar Multiplication | Multiplying every element of a matrix by a scalar value. | \( k \times \begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} ka & kb \\ kc & kd \end{bmatrix} \) |
| Identity Matrix (\( I \)) | A square matrix with 1s on the main diagonal and 0s elsewhere. Acts like the number 1 in matrix multiplication. | \( I_{2x2} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \) |
| Matrix Equality | Two matrices are equal if they have the same size and all corresponding elements are equal. | \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} e & f \\ g & h \end{bmatrix} \) implies \( a=e, b=f, c=g, d=h \) |
Additional Information: Properties of Matrix Transpose
The transpose operation has several useful properties in matrix algebra:
- \( (A')' = A \) (The transpose of the transpose of a matrix is the original matrix).
- \( (A + B)' = A' + B' \) (The transpose of a sum is the sum of the transposes, provided A and B have the same dimensions).
- \( (kA)' = kA' \) (The transpose of a scalar multiple is the scalar multiple of the transpose).
- \( (AB)' = B'A' \) (The transpose of a product is the product of the transposes in reverse order, provided the product AB is defined).
- For an identity matrix \( I \), \( I' = I \).
In this problem, we used the definition of the transpose and matrix addition, along with the property of matrix equality to solve for the variable \( x \).
Similar Questions
If \( A \) is a square matrix such that \( A^2 = 2I \), then the value of \( (A - I)^2 + (A + I)^2 - 7A \) is:
If \( A \) is a matrix and \( |A| \neq 0 \), then the solution of the equation \( XA = B \) is:
If \( A = \begin{bmatrix} 0 & -3 & 2 \\ a & b & 8 \\ -2 & c & 0 \end{bmatrix} \) is a skew-symmetric matrix, then:
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 :
