If \(\mathrm{A}=\left|\begin{array}{ll}\alpha & 0 \\ 2 & 3\end{array}\right|\) and \(\mathrm{I}=\left|\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right|\) then \(\mathrm{A}^{2}=91\) for (a) \(\alpha=4\) (b) \(\alpha=3\) (c) \(\alpha=-3\) (d) no \(\alpha\)

The determinant is a scalar value that can be computed from the elements of a square matrix. It provides important information about the matrix, such as whether the matrix is invertible and the volume scaling factor related to the linear transformations that the matrix represents.

Determinants have several key properties. For a 2x2 matrix \( A = \left|\begin{array}{ll}a & b \ c & d\end{array}\right| \), the determinant is calculated as \( ad - bc \). Determinants also play a crucial role in solving systems of linear equations and finding inverses of matrices. In our exercise, understanding determinants isn't directly required, but knowing that the determinant of the identity matrix (which has ones on its diagonal and zeros elsewhere) is 1 and its role in matrix equations could be informative.

An identity matrix, often denoted as \( I \), is a square matrix in which all the elements of the principal diagonal are ones, and all other elements are zeros. For instance, a 2x2 identity matrix looks like \( I = \left|\begin{array}{ll}1 & 0 \ 0 & 1\end{array}\right| \).

One of the key characteristics of the identity matrix is that when it's multiplied by any matrix of the same dimensions, it yields that matrix unchanged. So, \( AI = IA = A \) for any matrix \( A \). In our exercise, we compare the square of the matrix \( A \), \( A^2 \), to \( 91I \), making the identity matrix's properties quite relevant.

Matrix equations are equations where the variables are matrices. Solving such equations typically involves finding a matrix that satisfies the given conditions.

In the exercise, we're given the matrix equation \( A^2 = 91I \), where \( A \), a matrix with an unknown \( \alpha \), must be squared and then compared to \( 91 \) times the identity matrix. Understanding how to perform matrix multiplication and how to scale an identity matrix are both essential in solving this type of equation.

Algebraic operations on matrices, such as addition, subtraction, and multiplication, follow specific rules. For instance, matrices can only be added or subtracted if they have the same dimensions. Matrix multiplication, however, involves the 'dot product' of rows and columns and can occur if the number of columns in the first matrix matches the number of rows in the second matrix.

In our exercise, we need to multiply matrix \( A \) by itself to find \( A^2 \). The entries of the resulting matrix are determined by the sum of the products of the corresponding entries of the rows and columns from the original matrix \( A \)—a fundamental algebraic operation.

Solving matrix equations is a process that incorporates various algebraic techniques, including multiplication and comparison of matrices. To find a solution, you must often compare corresponding elements, exploit the properties of special matrices like the identity matrix, and use scalar multiplication.

In our problem, we solve the matrix equation by first squaring matrix \( A \), then scaling the identity matrix by 91, and finally comparing the resulting matrices to find the unknown \( \alpha \). We conclude that \( \alpha = -3 \) fits our equation, demonstrating a practical example of solving a matrix equation by equating corresponding elements and solving for the unknown.