Problem 71 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

a. \(a^2\), b. \(a^3\), c. \(a^4\), d. The pattern in the given determinants is that for the n x n matrix, all elements above the major diagonal are 'a' and all elements below are 0, e. The pattern in the evaluations is that the result for the determinant of a n x n matrix is \(a^n\).

Step 1: Evaluate 2x2 Determinant

Utilizing the determinant formula for a 2x2 matrix which is \((ad-bc)\), where matrix is \([a, b; c, d]\), evaluate \(|\begin{array}{ll}{a} & {a} \ {0} & {a}\end{array}|\). This results in \(a*a - a*0 = a^2\).

Step 2: Evaluate 3x3 Determinant

In this case, use the determinant formula for a 3x3 matrix. The determinant is \(|\begin{array}{ccc}{a} & {a} & {a} \ {0} & {a} & {a} \ {0} & {0} & {a}\end{array}|\), and applying the formula gives \(a*a*a - a*a*0 - a*0*a = a^3\).

Step 3: Evaluate 4x4 Determinant

Evaluating the 4x4 determinant \(|\begin{array}{cccc}{a} & {a} & {a} & {a} \ {0} & {a} & {a} & {a} \ {0} & {0} & {a} & {a} \ {0} & {0} & {0} & {a}\end{array}|\) will require using the standard method for calculating 4x4 determinants. This results in \(a*a*a*a - a*a*a*0 - a*a*0*a - a*0*a*a = a^4\).

Step 4: Identify the Pattern in the Determinants

In the given determinants, it can be observed that for the n x n matrix, all elements above the major diagonal are equal to 'a' and all elements below are 0.

Step 5: Identify the Pattern in the Evaluations

In the computed determinants, it can be observed that the answer for the determinant of a n x n matrix is \(a^n\), where n is the size of square matrix.

Key Concepts

2x2 Matrix Determinant

The determinant of a 2x2 matrix provides a value that can be used to calculate area, solve systems of equations, and predict the behavior of linear transformations. A 2x2 matrix has the form \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), and its determinant is calculated by the formula \(ad - bc\).

In the context of our exercise, the matrix \( \begin{bmatrix} a & a \ 0 & a \end{bmatrix} \) has its determinant evaluated as \(a \times a - a \times 0 = a^2\). This simplifies the process since any number multiplied by zero gives zero, and thus we don't need to include those terms in our calculation. The determinant in this case tells us that the area of the parallelogram (or the magnitude if you were thinking geometrically) defined by this matrix's columns is \(a^2\).

3x3 Matrix Determinant

Calculating the determinant of a 3x3 matrix is a step-up in complexity compared to a 2x2 matrix. Generally, for a matrix \( \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \) the determinant is given by \( aei + bfg + cdh - ceg - bdi - afh \).

In our example, we looked at a matrix where all non-diagonal, lower triangle elements were zero. This particular structure simplifies to \(a \times a \times a = a^3\), showing a pattern that each determinant is the product of the diagonal elements only. This reveals a useful shortcut for finding determinants of matrices with certain structures.

4x4 Matrix Determinant

The 4x4 matrix is even more complex, and its determinant is composed of multiple parts derived from 3x3 matrices. Yet, we encounter a special case in our exercise: a triangular matrix, where all elements below the main diagonal are zeros. The determinant for a 4x4 matrix like this is simply the product of the diagonal elements. So, \( \begin{bmatrix} a & a & a & a \ 0 & a & a & a \ 0 & 0 & a & a \ 0 & 0 & 0 & a \end{bmatrix} \) gives us \(a \times a \times a \times a = a^4\).

Taking this into consideration, understanding special cases, such as those with many zeros, can vastly simplify determinant calculation and offers a glimpse into the beauty and patterns of algebra.

Pattern Recognition in Algebra

Pattern recognition in algebra entails identifying regularities and using them to solve problems more efficiently. Here, noticing that each matrix is triangular (i.e., all elements below the main diagonal are zero) allowed us to quickly find the determinant, sidestepping more complex calculations.

Key Observations:

• All matrices in the exercise are square and special types known as 'upper triangular'.
• The determinant of an upper triangular matrix is the product of its diagonal elements.
• For these matrices, the determinant thus follows the pattern \(a^n\), where \(n\) is the size of the matrix.

Recognizing and applying such patterns can save time and help understand deeper algebraic concepts.