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The image asks to determine the values of \( a \) for which the given matrix is orthogonal. The matrix is:

\[
A = \begin{bmatrix} 5 & 6 \\ 1 & a \end{bmatrix}
\]

### To determine when a matrix is orthogonal:

A matrix \( A \) is orthogonal if its transpose is equal to its inverse:

\[
A^T A = I
\]

where \( I \) is the identity matrix.

For the given matrix:

\[
A = \begin{bmatrix} 5 & 6 \\ 1 & a \end{bmatrix}, \quad A^T = \begin{bmatrix} 5 & 1 \\ 6 & a \end{bmatrix}
\]

Now compute \( A^T A \):

\[
A^T A = \begin{bmatrix} 5 & 1 \\ 6 & a \end{bmatrix} \begin{bmatrix} 5 & 6 \\ 1 & a \end{bmatrix}
= \begin{bmatrix} 5 \cdot 5 + 1 \cdot 1 & 5 \cdot 6 + 1 \cdot a \\ 6 \cdot 5 + a \cdot 1 & 6 \cdot 6 + a \cdot a \end{bmatrix}
\]

Simplify the multiplication:

\[
A^T A = \begin{bmatrix} 25 + 1 & 30 + a \\ 30 + a & 36 + a^2 \end{bmatrix}
= \begin{bmatrix} 26 & 30 + a \\ 30 + a & 36 + a^2 \end{bmatrix}
\]

For \( A \) to be orthogonal, we need:

\[
A^T A = I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
\]

This gives us two equations:

1. \( 26 = 1 \)
2. \( 30 + a = 0 \)
3. \( 36 + a^2 = 1 \)

The first equation is not possible, so no value of \( a \) will make this matrix orthogonal. Therefore, matrix \( A \) is not orthogonal for any real number value of \( a \).

Would you like more details, or do you have any questions?

Here are some related questions you might be interested in:
1. What is an orthogonal matrix and how is it used?
2. How do you find the inverse of a 2x2 matrix?
3. What are some properties of orthogonal matrices?
4. What is the geometric interpretation of an orthogonal matrix?
5. How do orthogonal transformations relate to linear algebra?

**Tip:** Remember that an orthogonal matrix represents a rotation or reflection in space.

Determine the value of a for which the matrix A = [[5, 6], [1, a]] is orthogonal.

Explore how to find the value of 'a' that makes the matrix A = [[5, 6], [1, a]] orthogonal. Understand the concept of orthogonal matrices and the criteria for matrix orthogonality through step-by-step calculations and formulas. Suitable for undergraduate students learning linear algebra.

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