If `A=[(x,0),(1,1)]andB=[(1,0),(5,1)]` find x such that `A^(2)=B`.

Answer

To solve the problem, we need to find the value of \( x \) such that \( A^2 = B \), where

\[ A = \begin{pmatrix} x & 0 \\ 1 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & 0 \\ 5 & 1 \end{pmatrix}. \]

### Step 1: Calculate \( A^2 \)

To find \( A^2 \), we multiply matrix \( A \) by itself:

\[ A^2 = A \cdot A = \begin{pmatrix} x & 0 \\ 1 & 1 \end{pmatrix} \cdot \begin{pmatrix} x & 0 \\ 1 & 1 \end{pmatrix}. \]

### Step 2: Perform the matrix multiplication

Using the formula for matrix multiplication, we calculate each element of the resulting matrix:

- The element at the first row, first column:
\[ x \cdot x + 0 \cdot 1 = x^2. \]

- The element at the first row, second column:
\[ x \cdot 0 + 0 \cdot 1 = 0. \]

- The element at the second row, first column:
\[ 1 \cdot x + 1 \cdot 1 = x + 1. \]

- The element at the second row, second column:
\[ 1 \cdot 0 + 1 \cdot 1 = 1. \]

Thus, we have:

\[ A^2 = \begin{pmatrix} x^2 & 0 \\ x + 1 & 1 \end{pmatrix}. \]

### Step 3: Set \( A^2 \) equal to \( B \)

Now we set \( A^2 \) equal to \( B \):

\[ \begin{pmatrix} x^2 & 0 \\ x + 1 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 5 & 1 \end{pmatrix}. \]

### Step 4: Create equations from the matrix equality

From the equality of the matrices, we can create the following equations:

1. \( x^2 = 1 \)
2. \( 0 = 0 \) (This is always true and gives no information)
3. \( x + 1 = 5 \)
4. \( 1 = 1 \) (This is also always true)

### Step 5: Solve the equations

From the first equation \( x^2 = 1 \):

\[ x = 1 \quad \text{or} \quad x = -1. \]

From the third equation \( x + 1 = 5 \):

\[ x = 5 - 1 = 4. \]

### Step 6: Check for consistency

We have two possible values for \( x \) from the first equation: \( x = 1 \) and \( x = -1 \). However, from the third equation, we found \( x = 4 \).

Since we have conflicting values for \( x \), we conclude that there is no single value of \( x \) that satisfies both equations.

### Final Conclusion

Thus, there is no such \( x \) that satisfies \( A^2 = B \).

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