Question

Let $$ A=\left|\begin{array}{ccc} 4 & 4 \mathrm{k} & \mathrm{k} \\ 0 & \mathrm{k} & 4 \mathrm{k} \\ 0 & 0 & 4 \end{array}\right| $$ If \(\operatorname{det}\left(\mathrm{A}^{2}\right)=16\) then \(|\mathrm{k}|\) is \(\ldots \ldots\) (a) 1 (b) \((1 / 4)\) (c) 4 (d) \(4^{2}\)

Short answer

The absolute value of k is 1. The correct answer is (a) 1.

Step-by-step solution

Determine the determinant of matrix A

Since the matrix A is an upper triangular matrix, we can find the determinant by simply multiplying the diagonal entries. Let δ(A) denote the determinant of A: \( δ(A) = 4 × k × 4 \)

Square the determinant of A

Now we'll square the determinant of A to match the given condition that the determinant of A squared is equal to 16. Let's denote the determinant of A squared as δ(A^2): \( δ(A^2) = (4 × k × 4)^2 \)

Set the determinant of A squared to 16

Since the problem states that the determinant of A squared is equal to 16, we can plug in the determinant found in step 2 to form the equation: \( (4 × k × 4)^2 = 16 \)

Solve for k

Now we just need to solve this equation to find the absolute value of k: \( (16 × k^2) = 16 \) \( k^2 = 1 \) Taking the square root on both sides, we are left with: \( k = ±1 \) So the absolute value of k is 1. The correct answer is (a) 1.

Key concepts

Upper Triangular Matrix

Understanding the concept of an upper triangular matrix is crucial for students delving into matrix algebra. An upper triangular matrix is a special form of square matrix where all the elements below the main diagonal are zero. The main diagonal itself can have non-zero elements. Visually, any numbers that would appear on the left side of the diagonal from the top-left to the bottom-right of the matrix are not there – they're just zeroes.

An important property of an upper triangular matrix is that its determinant is simply the product of its diagonal entries. This means that to find the determinant of such a matrix, we do not need to perform any complex operations; we just multiply the numbers in the diagonal. When a student encounters a large and complex matrix, identifying whether it is upper triangular can vastly simplify the process of finding its determinant. This comes in handy when working with matrix equations where the determinant is a necessary value.

Matrix Algebra

Matrix algebra is a cornerstone of linear algebra and has vast applications in various fields including engineering, physics, computer science, and economics. It involves operations like addition, subtraction, multiplication, and finding the determinant of matrices. Unlike regular algebra, these operations follow specific rules pertaining to the dimensions and structure of matrices.

One of the fundamental operations in matrix algebra is determining the determinant of a matrix, which can be interpreted as a value that gives information about the matrix's characteristics, such as whether it has an inverse or its transformation properties. The concept of squaring a matrix, seen in our exercise with the term 'determinant of A squared', involves finding the determinant of a matrix that has been multiplied by itself. Students should note that squaring a determinant is not the same as squaring a matrix. These nuances are vital to solving matrix algebra problems accurately.

Absolute Value

The absolute value of a number represents its distance from zero on the number line, regardless of its direction. Symbolically, it's represented by two vertical bars surrounding the number, for example, \( |k| \). The absolute value ignores whether the number is positive or negative; hence, it is always a non-negative value.

When dealing with equations in algebra that involve absolute values, it's important to remember that there can be two possible solutions. This is because both positive and negative versions of a number have the same absolute value. In the context of our textbook exercise, the variable \( k \) could potentially be positive or negative, but its absolute value \( |k| \) yields a positive result. Solving for \( k \) involves considering both possibilities and then determining the absolute value. Understanding this concept ensures that students do not miss alternative solutions to problems involving absolute values.