Problem 58 - Algebra 2

Question

Evaluate the determinant of each matrix. $$ \left[\begin{array}{rrr}{6} & {5} & {2} \\ {-3} & {0} & {-6} \\ {1} & {4} & {2}\end{array}\right] $$

Step-by-Step Solution

Answer

The determinant is 120.

Step 1: Understand the Determinant Formula for a 3x3 Matrix

The determinant of a 3x3 matrix \(\begin{bmatrix} a & b & c \d & e & f \g & h & i \\end{bmatrix}\) is given by the formula:\[ ext{det} = a(ei − fh) − b(di − fg) + c(dh − eg).\] This formula involves calculating the determinants of the 2x2 matrices formed by eliminating one row and one column from the 3x3 matrix.

Step 2: Identify Matrix Elements

For the matrix \(\begin{bmatrix} 6 & 5 & 2 \-3 & 0 & -6 \1 & 4 & 2 \end{bmatrix},\)assign the variables as follows: \(a=6\), \(b=5\), \(c=2\), \(d=-3\), \(e=0\), \(f=-6\), \(g=1\), \(h=4\), and \(i=2\).

Step 3: Calculate the Minor Determinants

Calculate the minors for the determinant formula:1. \(ei - fh = (0 \cdot 2) - (-6 \cdot 4) = 0 + 24 = 24\).2. \(di - fg = (-3 \cdot 2) - (-6 \cdot 1) = -6 + 6 = 0\).3. \(dh - eg = (-3 \cdot 4) - (0 \cdot 1) = -12 + 0 = -12\).

Step 4: Substitute Back into the Determinant Formula

Plug the values from the minor calculations into the determinant formula:\[ ext{det} = 6(24) - 5(0) + 2(-12).\]This simplifies to \(6 \cdot 24 + 0 + 2 \cdot (-12)\).

Step 5: Evaluate the Final Expression

Carry out the multiplication and addition:1. \( 6 \cdot 24 = 144\).2. \( 5 \cdot 0 = 0\).3. \( 2 \cdot (-12) = -24\).Combine these results: \(144 + 0 - 24 = 120\).

Step 6: Conclusion

The determinant of the matrix is \(120\). Therefore, the matrix is invertible and non-degenerate since its determinant is a non-zero value.

Key Concepts

Matrix Inversion

Matrix inversion is the process of finding a matrix that, when multiplied with the original square matrix, results in the identity matrix. Think of the identity matrix as the equivalent of '1' in matrix terms. Only square matrices, specifically those that are non-degenerate, can have inverses. This means the determinant must be non-zero.

• To find the inverse of a non-degenerate 3x3 matrix, the determinant of the matrix must be calculated as the first step.
• Next, we need to find the matrix of minors, then convert it to the matrix of cofactors and transpile it to create the adjugate matrix.
• Finally, the inverse is obtained by dividing the adjugate matrix by the original matrix's determinant.

Remember, if the determinant is zero, the inverse does not exist. Hence, calculating the determinant as we did is crucial for inversion.

2x2 Matrix Determinants

The concept of 2x2 determinants is essential when dealing with larger matrices like 3x3. Each term in the 3x3 determinant formula involves a 2x2 determinant.

• A 2x2 matrix is structured like: \[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \].
• The determinant for this matrix is calculated as: \[ ad - bc \].

The 3x3 matrix determinants rely heavily on this formula. Each minor determinant you calculate comes from omitting one row and one column, then evaluating the remaining 2x2 matrix.
This fundamental understanding simplifies more complex matrices, such as 3x3.

Minor Determinants

Minor determinants are the "building blocks" for calculating larger matrix determinants. In a 3x3 matrix, a minor is a 2x2 determinant formed by removing one row and one column from the original matrix.

• The minor for each element involves identifying the 2x2 matrix that remains when the row and column of that element are omitted.
• For instance, in our example, the minor determinant for the element at position (1, 1) is \[ (0 \cdot 2) - (-6 \cdot 4) \].The minor determinant helps construct the complete determinant for the matrix.

Understanding minor determinants is critical as they simplify calculations for larger matrices. Use the consistent pattern of eliminating a row and column to find these minor determinants effectively.