In Exercises 57–60, solve each equation for x.

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Step 1: Recognize that the problem involves finding the value of \( x \) such that the determinant of the given 3x3 matrix equals \( -8 \). The matrix is \( \begin{bmatrix} 1 & x & -2 \\ 3 & 1 & 1 \\ 0 & -2 & 2 \end{bmatrix} \).

Step 2: Recall the formula for the determinant of a 3x3 matrix \( A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \) is \( \det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \).

Step 3: Apply the determinant formula to the given matrix, substituting \( a=1 \), \( b=x \), \( c=-2 \), \( d=3 \), \( e=1 \), \( f=1 \), \( g=0 \), \( h=-2 \), and \( i=2 \).

Step 4: Calculate each minor determinant: \( ei - fh = (1)(2) - (1)(-2) \), \( di - fg = (3)(2) - (1)(0) \), and \( dh - eg = (3)(-2) - (1)(0) \).

Step 5: Substitute these values back into the determinant expression, simplify the resulting equation, and set it equal to \( -8 \). Then solve the resulting linear equation for \( x \).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Determinant of a 3x3 Matrix

The determinant is a scalar value that can be computed from a square matrix and provides important properties such as invertibility. For a 3x3 matrix, the determinant is calculated using expansion by minors or the rule of Sarrus, involving sums and products of matrix elements. Understanding how to compute this determinant is essential to solving the given equation.

Solving Equations Involving Variables in Matrices

When a variable appears inside a matrix, such as 'x' in this problem, the determinant becomes an expression in terms of that variable. Solving the equation involves setting the determinant equal to the given value and solving the resulting algebraic equation for x. This requires combining matrix operations with algebraic manipulation.

Properties of Determinants and Their Applications

Determinants have properties like linearity and the effect of row operations, which can simplify calculations. Recognizing these properties helps in efficiently computing the determinant and solving for variables. Additionally, the determinant's value can indicate matrix invertibility, which is often relevant in algebra problems.