Problem 38 - Algebra and Trigonometry

Question

Evaluate each determinant. $$\left|\begin{array}{rrrr}{3} & {-1} & {1} & {2} \\\\{-2} & {0} & {0} & {0} \\\\{2} & {-1} & {-2} & {3} \\\\{1} & {4} & {2} & {3}\end{array}\right|$$

Step-by-Step Solution

Answer

The determinant of the given 4x4 matrix is 4

Step 1: Create smaller matrices

First, let's make four 3x3 matrices by choosing a row or column and evaluate them. Typically we choose the row or column with the maximum number of zeros to simplify calculations. Here, we can select the second row since it contains three zeroes. So, our four smaller matrices are: \[M_1 = \left| \begin{array}{ccc} -1 & 1 & 2 \ -1 & -2 & 3 \ 4 & 2 & 3 \end{array} \right|, M_2 = \left| \begin{array}{ccc} 3 & 1 & 2 \ 2 & -2 & 3 \ 1 & 2 & 3 \end{array} \right|,M_3 = \left| \begin{array}{ccc} 3 & -1 & 2 \ 2 & -1 & 3 \ 1 & 4 & 3 \end{array} \right|M_4 = \left| \begin{array}{ccc} 3 & -1 & 1 \ 2 & -1 & -2 \ 1 & 4 & 2 \end{array} \right|\]

Step 2: Calculate the determinants of the smaller matrices

Next, we calculate the determinants of each of these 3x3 matrices using the rule of Sarrus:\[| M_1| = -1*(-2*3 - 3*2) - 1*(2*3 - 3*4) + 2*(-1*4 - 1*2) = -2,| M_2 | = 3*(-2*3 - 3*2) - 1*(2*3 - 3*1) + 2*(1*4 - 2*2) = -4,| M_3 | = 3*(-1*3 - 3*2) - (-1)*(2*3 - 3*1) + 2*(1*4 - 2*2) = 12,| M_4 | = 3*(-1*3 - 2*2) - (-1)*(2*2 - 3*1) + 1 *(1*4 - 2*2) = 7\]

Step 3: Combine the determinants of the smaller matrices

Next, combine these determinants according to the formula for the determinant of a 4x4 matrix based on the selected row, taking into account the sign. The formula is:\[| A | = a_{ij} * (-1)^{i+j} * | M_{ij} |Let's compute:\[| A | = (-2) * (-1)^{2+1} * -2 + 0 * (-1)^{2+2} * -4 + 0 * (-1)^{2+3} * 12 + 0 * (-1)^{2+4} * 7 \]

Key Concepts

Determinant of a Matrix

The determinant of a matrix is a value that can be computed from a square matrix. It is a unique number, which is useful in analysis and solution of linear equations, finding inverse of matrices and in calculus. When we look at a 2x2 matrix, for example \[A = \left[\begin{array}{cc} a & b \ c & d \end{array}\right]\], the determinant is calculated as \(ad - bc\).For larger square matrices, such as 3x3 or 4x4, the process to evaluate the determinant gets more complex. This is where we might use techniques like the rule of Sarrus for 3x3 matrices or cofactor expansion for any nxn matrices. When evaluating the determinant, it's crucial to look for patterns or rows and columns which may simplify the process, such as those containing zeros.

Rule of Sarrus

The rule of Sarrus is a technique used to calculate the determinant of a 3x3 matrix. It's a simple method where you write down the matrix and repeat its first two columns on the right. Then, you sum the products of the diagonals that go from the upper-left to the lower-right and subtract the sum of the products of the diagonals that go from the lower-left to the upper-right. So, for a matrix \[A = \left[\begin{array}{ccc} a & b & c \ d & e & f \ g & h & i \end{array}\right]\], we calculate the determinant as \((aei + bfg + cdh) - (ceg + bdi + afh)\). This method is efficient for 3x3 matrices but doesn't extend to larger matrices. For 4x4 matrices or bigger, you would use the method of cofactor expansion instead.

Submatrices in Determinants

When dealing with a matrix larger than 3x3, we need to break it down into smaller parts, known as submatrices, to calculate its determinant. This involves selecting a row or column (preferably with the most zeros to simplify calculations) and removing it to create smaller square matrices. Each of these submatrices helps us in using the cofactor expansion method to find the determinant of the original matrix. The determinants of these submatrices play a critical role, as they are multiplied by their respective cofactor and contribute to the overall determinant. In the given exercise, the second row was chosen for its abundance of zeroes to create four 3x3 submatrices.

Cofactor Expansion

Cofactor expansion, also known as Laplace's expansion, is a procedure to calculate the determinant of any square matrix. We expand along a row or column, generally choosing one with the most zeros to simplify the calculation, and for each element in the row or column, we find the determinant of the matrix that remains after removing the element's row and column. This 'smaller determinant' is then multiplied by the original element and its sign, given by \( (-1)^{i+j} \), where \(i\) and \(j\) are the row and column numbers of the element. The sum of these products gives us the determinant of the original matrix. For instance, in a 4x4 matrix, we would compute the determinant as \(a_{ij} (-1)^{i+j} |M_{ij}|\), where \(a_{ij}\) is an element from the original matrix, and \(M_{ij}\) is the determinant of its submatrix. The process combines these smaller determinants using rules of arithmetic and the proper signs to reach the final determinant.