91Ó°ÊÓ

The determinant of a matrix helps us understand certain properties of the matrix, including whether it's invertible. For a 3x3 matrix, the determinant can be calculated using a specific formula. The formula uses the elements of the matrix in a way that involves multiplying and summing up products of different elements. This calculation can tell us if a matrix is invertible or not, based on whether the determinant is zero or non-zero. In our problem, finding the determinant of matrix \( \boldsymbol{A} \) and setting it to zero helped us find potential values of \( a \) for which the matrix may not be invertible. However, we discovered that there is no real value of \( a \) that makes \( \boldsymbol{A} \) non-invertible, since the determinant equation resulted in a negative discriminant, indicating there are no real roots. Understanding this process is essential, as knowing when a matrix is invertible or not is crucial in solving systems of equations.

A system of linear equations is essentially a collection of two or more linear equations involving the same set of variables. Solving such a system often involves finding values for the variables that satisfy all equations simultaneously. There are several techniques to solve these systems, including substitution, elimination, and matrix methods. In the context of matrices, if the matrix is invertible, we can use its inverse to find the solution. That is precisely what we did in this exercise. Given the matrix equation \( \boldsymbol{A} \boldsymbol{x} = \begin{bmatrix} 1 \ 0 \ 2 \end{bmatrix} \), we found the inverse of the matrix \( \boldsymbol{A} \) when \( a=2 \), and then multiplied this inverse by the vector on the right. By doing so, we found the solution vector \( \boldsymbol{x} \), giving us the values of \(x\), \(y\), and \(z\). This method is very powerful, especially when dealing with complex systems involving multiple variables.

A quadratic equation is a second-degree polynomial equation of the form \( ax^2 + bx + c = 0 \). Solving quadratic equations is crucial in various fields including mathematics, physics, and engineering. The quadratic formula is a well-known method for finding the roots of such equations: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). In our exercise, when attempting to find real values for \( a \) making the determinant zero, we applied the quadratic formula to the equation \( a^2 + 2a + 2 = 0 \). However, the discriminant, given by \( b^2 - 4ac \), was negative in this case, resulting in no real solutions. This highlights an important aspect of quadratic equations: the nature of the roots, whether real or complex, is determined by the sign of the discriminant. Understanding how to interpret the discriminant allows us to anticipate the type of solutions we might encounter.