91Ó°ÊÓ

When studying linear algebra, one important concept is the \( determinant \) of a matrix. This single number provides valuable information about the matrix, such as whether a system of linear equations has a unique solution or if the matrix is invertible. In the context of a 2x2 matrix, the determinant is calculated by subtracting the product of the elements on the anti-diagonal from the product of the diagonal elements. For example, if we have a matrix:
\[ \begin{pmatrix} a & b \ c & d \end{pmatrix} \],
then the determinant (denoted as \( det \)) is calculated as \( ad - bc \).

In the given exercise, we are presented with a matrix where \( a = x \) and \( d = x \), giving us the determinant equation \( x^2 - 2*1 \), or \( x^2 - 2 \). Calculating the determinant allows us to proceed with solving the absolute value equation given.

Absolute value equations involve expressions within absolute value symbols, which denote the distance of a number from zero on the number line, regardless of direction. For example, \( |x| = 3 \) has two solutions: \( x = 3 \) and \( x = -3 \).

In our exercise, the absolute value of the matrix's determinant—\( |x^2 - 2| \)—is set equal to 2. This means we must consider both positive and negative scenarios: \( x^2 - 2 = 2 \) and \( x^2 - 2 = -2 \). Solving these two separate equations will give us all possible values of \( x \) that satisfy the original absolute value statement. It's crucial when solving such problems to remember to account for both scenarios, as missing either can lead to incomplete solutions.

Quadratic equations, which are of the form \( ax^2 + bx + c = 0 \), are fundamental in algebra. They are characterized by the highest exponent being two. These equations can have two real solutions, one real solution, or two complex solutions. The solutions can be found by factoring, completing the square, using the quadratic formula, or—when possible—extracting square roots.

In the exercise, the equations resulting from the determinant and absolute value simplification are basic quadratics: \( x^2 - 2 = 2 \) and \( x^2 - 2 = -2 \). By rearranging them to standard quadratic form, we get \( x^2 - 4 = 0 \) and \( x^2 = 2 \), respectively. Solving these gives the roots \( x = 2, -2, 0 \), demonstrating a practical application of quadratic equations in conjunction with other mathematical concepts.