Problem 63 - College Algebra

Question

a. Evaluate: \(\left|\begin{array}{ll}a & a \\ 0 & a\end{array}\right|\) b. Evaluate: \(\left|\begin{array}{lll}a & a & a \\ 0 & a & a \\ 0 & 0 & a\end{array}\right|\) c. Evaluate: \(\left|\begin{array}{llll}a & a & a & a \\ 0 & a & a & a \\ 0 & 0 & a & a \\ 0 & 0 & 0 & a\end{array}\right|\) d. Describe the pattern in the given determinants. e. Describe the pattern in the evaluations.

Step-by-Step Solution

Answer

The evaluations of the determinants are \(a^2\), \(a^3\), and \(a^4\) respectively. The pattern in the determinants is having all 'a' in the main diagonal and the positions above, with the rest being zeros. The evaluation results in \(a^n\), where n is the matrix dimension.

Step 1: Evaluate 2x2 Matrix

The determinant of a 2x2 matrix, \(\left|\begin{array}{ll}a & b \ c & d\end{array}\right|\), is calculated as ad - bc. In this case \(\left|\begin{array}{ll}a & a \ 0 & a\end{array}\right|\), the determinant is a*a - a*0, which simplifies to \(a^2\).

Step 2: Evaluate 3x3 Matrix

The determinant of a 3x3 matrix, \(\left|\begin{array}{lll}a & b & c \ d & e & f \ g & h & i\end{array}\right|\), is calculated as a(ei-fh) - b(di-fg) + c(dh-eg). However, because of the zeros in this problem, many terms in this formula drop out due to multiplication with zero. This leaves the determinant for this matrix \(\left|\begin{array}{lll}a & a & a \ 0 & a & a \ 0 & 0 & a\end{array}\right|\) as a * a * a - 0 - 0 + 0 - 0 + 0. This simplifies to \(a^3\).

Step 3: Evaluate 4x4 Matrix

A 4x4 determinant calculation can be quite complex. However, in this case, thanks to the matrix's structure, the calculation is significantly simplified. The result for this 4x4 matrix \(\left|\begin{array}{llll}a & a & a & a \ 0 & a & a & a \ 0 & 0 & a & a \ 0 & 0 & 0 & a\end{array}\right|\) is \(a^4\).

Step 4: Describe the Pattern in the Determinants

Looking at the matrices, there is clearly a pattern. The main diagonal (top-left to bottom-right) has all 'a' and the positions above the main diagonal also have 'a', the remaining, below the main diagonal, are all zero.

Step 5: Describe the Pattern in the Evaluation

In each matrix, the determinant amounts to \(a^n\), where n is the dimension of the square matrix. This is directly due to the zeros in the lower triangle of the matrix simplifying the calculation.