Problem 69 - Precalculus

Question

Explain the mistake that is made. Evaluate the determinant \(\left|\begin{array}{rrr}2 & 1 & 3 \\ -3 & 0 & 2 \\\ 1 & 4 & -1\end{array}\right|\) Solution: Expand the \(3 \times 3\) determinant in terms of the \(2 \times 2\) determinants. \(\left|\begin{array}{rrr}2 & 1 & 3 \\ -3 & 0 & 2 \\ 1 & 4 & -1\end{array}\right|=2\left|\begin{array}{rr}0 & 2 \\ 4 & -1\end{array}\right|+1\left|\begin{array}{rr}-3 & 2 \\ 1 & -1\end{array}\right|+3\left|\begin{array}{rr}-3 & 0 \\ 1 & 4\end{array}\right|\) Expand the \(2 \times 2\) determinants. \(\quad=2(0-8)+1(3-2)+3(-12-0)\) Simplify. \(=-16+1-36=-51\) This is incorrect. What mistake was made?

Step-by-Step Solution

Answer

The error was in the incorrect calculation of the determinant, which should be \(-53\), not \(-51\).

Step 1: Reviewing the Expansion Formula for 3x3 Determinant

The determinant of a 3x3 matrix \( \left| \begin{array}{ccc} a & b & c \ d & e & f \ g & h & i \end{array} \right| \) is given by \( a(ei-fh) - b(di-fg) + c(dh-eg) \). This step involves using co-factors of the first row of the matrix, matching the sign pattern of the determinant's minors: \( (+, -, +) \).

Step 2: Applying the Correct Formula to the Given Matrix

The matrix is \( \left| \begin{array}{ccc} 2 & 1 & 3 \ -3 & 0 & 2 \ 1 & 4 & -1 \end{array} \right| \). According to the formula, the expression becomes:\(2((0)(-1) - (2)(4)) - 1((-3)(-1) - (2)(1)) + 3((-3)(4) - (0)(1))\).

Step 3: Calculating Each 2x2 Determinant

Calculate the determinants of the 2x2 matrices:- \((0)(-1) - (2)(4) = 0 - 8 = -8\)- \((-3)(-1) - (2)(1) = 3 - 2 = 1\)- \((-3)(4) - (0)(1) = -12 - 0 = -12\)

Step 4: Combine the Results Correctly

Using the formula:\(2(-8) - 1(1) + 3(-12)\) simplifies to \(-16 - 1 - 36\).

Step 5: Confirm Total Calculation

Perform the final calculation:\(-16 - 1 - 36 = -53\).This shows the original final value given in the provided solution was incorrect.

Key Concepts

3x3 Matrix

When we talk about a 3x3 matrix in linear algebra, we refer to a matrix that consists of three rows and three columns. Each entry within the matrix holds a numerical value. The matrix looks like this:\[\begin{array}{ccc}a & b & c \d & e & f \g & h & i \\end{array}\]Understanding how to work with a 3x3 matrix is essential, especially when you want to calculate its determinant. The determinant is a special number that can help in solving systems of linear equations, among other applications.
To compute the determinant of a 3x3 matrix, we generally use the method of cofactor expansion, which involves breaking down the matrix into smaller 2x2 matrices and calculating their determinants. This allows us to manage the calculation more efficiently.

Cofactor Expansion

Cofactor expansion is a technique used to calculate the determinant of a matrix, especially useful for matrices larger than 2x2, such as a 3x3 matrix. The method picks one row or one column of the matrix to "expand" across, often choosing the first row for simplicity.For instance:- Select the first row of the matrix: \(a, b, c\).- Use the sign pattern for determinants: \((+,-,+)\).The formula for the determinant is:\[a(ei - fh) - b(di - fg) + c(dh - eg)\]Each term like \(a(ei - fh)\) is the product of:- The matrix's entry \(a\),- The determinant of a 2x2 matrix formed by eliminating the row and column of \(a\),- And the proper sign "+" or "-" from the sequence.Understanding why cofactor expansion works may involve deeper learning about linear algebra concepts such as linear transformations and their properties, but as a basic rule, it helps calculate determinants efficiently. Always remember the prominent sign pattern, as neglecting it can lead to incorrect answers.

2x2 Determinants

2x2 determinants are fundamental building blocks used when we calculate the determinant of larger matrices, like a 3x3 matrix. The 2x2 determinant is relatively straightforward to calculate and is given by the expression:\[\begin{vmatrix} a & b \c & d \end{vmatrix} = ad - bc\]This expression reflects the diagonal rule, where you multiply the values of one diagonal \((ad)\) and subtract the product of the other diagonal \((bc)\).
In a 3x3 matrix, when using cofactor expansion, we frequently encounter scenarios where we have to determine the value of these 2x2 sub-matrices. Therefore, fluency in handling 2x2 determinants is crucial.Whenever you expand a 3x3 matrix into 2x2 matrices, be sure to correctly identify each smaller matrix's elements. Misidentifying these leads to errors, as we saw in the original step-by-step solution where recalculation of each minor was needed to rectify mistakes.