If you have worked steadily through the Programme, you...

If you have worked steadily through the Programme, you should have no difficulty with this exercise. Take your time and work carefully. There is no extra credit for speed. Off you go then. They are all quite straightforward. Solve the equation: $$ \left|\begin{array}{ccc} x+1 & -5 & -6 \\ -1 & x & 2 \\ -3 & 2 & x+1 \end{array}\right|=0 $$

If you have worked steadily through the Programme, you should have no difficulty with this exercise. Take your time and work carefully. There is no extra credit for speed. Off you go then. They are all quite straightforward.
Solve the equation:
$$
\left|\begin{array}{ccc}
x+1 & -5 & -6 \\
-1 & x & 2 \\
-3 & 2 & x+1
\end{array}\right|=0
$$

Key Concepts

Determinant

The determinant is a scalar value computed from a square matrix that encapsulates important properties of the matrix system, such as invertibility and volume scaling. It is also used in solving systems of linear equations and in finding eigenvalues. Calculating the determinant of a matrix provides insight into whether the matrix is singular (non-invertible) when the determinant equals zero.

Cofactor Expansion

Cofactor expansion is a method used to compute the determinant of a matrix. This technique involves selecting a row or column, then calculating the sum of the products of each element with its corresponding cofactor, where each cofactor is the determinant of the submatrix formed by removing the element's row and column, multiplied by a sign factor. This method is particularly useful for small matrices or when the matrix has zeros that simplify the calculation.

Polynomial Equations

When an expression, such as a determinant, is set equal to zero, it often results in a polynomial equation in one or more variables. Solving such equations involves factoring the polynomial, using methods like the quadratic formula, synthetic division, or the Rational Root Theorem. This process finds the values of the variable that satisfy the original determinant equation.

Matrix Algebra

Matrix algebra encompasses the operations and properties of matrices, including addition, multiplication, and the computation of determinants. It provides the foundational tools to work with systems of linear equations, transformations, and other advanced topics in linear algebra. Understanding matrix algebra is essential for analyzing how changes in a parameter, like x in a determinant, affect the properties of the matrix.

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