Questions: Now find the following: A · A^t = [[a b] [c d]] where A^t is the transform of A. a=0, b=0, b, c=0

Solution Steps

To solve the given problem, we need to compute the product of matrix \( A \) and its transpose \( A^t \). The transpose of a matrix is obtained by swapping its rows and columns. Once we have the transpose, we can perform matrix multiplication to find the resulting matrix.

1. Define the matrix \( A \).
2. Compute the transpose of \( A \).
3. Perform matrix multiplication of \( A \) and \( A^t \).
4. Extract the elements \( a, b, c, d \) from the resulting matrix.

We are given a matrix \( A \) and its transpose \( A^t \). We need to find the product \( A \cdot A^t \) and identify the resulting matrix. The problem also provides specific values for the elements of the resulting matrix.

Let's assume \( A \) is a 2x2 matrix: \[ A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} \]

The transpose of \( A \), denoted as \( A^t \), is: \[ A^t = \begin{bmatrix} a_{11} & a_{21} \\ a_{12} & a_{22} \end{bmatrix} \]

The product \( A \cdot A^t \) is calculated as follows: \[ A \cdot A^t = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} \cdot \begin{bmatrix} a_{11} & a_{21} \\ a_{12} & a_{22} \end{bmatrix} \]

\[ A \cdot A^t = \begin{bmatrix} a_{11}a_{11} + a_{12}a_{12} & a_{11}a_{21} + a_{12}a_{22} \\ a_{21}a_{11} + a_{22}a_{12} & a_{21}a_{21} + a_{22}a_{22} \end{bmatrix} \]

We are given: \[ \left[\begin{array}{ll} a & b \\ c & d \end{array}\right] = \left[\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right] \]

This implies: \[ \begin{cases} a_{11}a_{11} + a_{12}a_{12} = 0 \\ a_{11}a_{21} + a_{12}a_{22} = 0 \\ a_{21}a_{11} + a_{22}a_{12} = 0 \\ a_{21}a_{21} + a_{22}a_{22} = 0 \end{cases} \]

From the equations, we can infer that: \[ a_{11}^2 + a_{12}^2 = 0 \\ a_{21}^2 + a_{22}^2 = 0 \]

Since the sum of squares of real numbers is zero, each individual term must be zero: \[ a_{11} = 0, \quad a_{12} = 0, \quad a_{21} = 0, \quad a_{22} = 0 \]

Given that all elements of \( A \) are zero, the resulting matrix \( A \cdot A^t \) is: \[ A \cdot A^t = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \]

The resulting matrix is a zero matrix.

Final Answer