Problem 96 - Precalculus with Limits

Question

THINK ABOUT IT If \(A\) is a matrix of order \(3 \times 3\) such that \(|A|=5\), is it possible to find \(|2A|\)? Explain.

Step-by-Step Solution

Answer

Yes, it is possible to find |2A|. The determinant of the matrix 2A, i.e., |2A|, is 40.

Step 1: Review of Property

Understand the property of determinant. For any square matrix A of order n and any scalar k, |kA| = \(k^n |A|\). The order (n) is the dimension of the matrix, in this case 3. So it means, if we multiply all entries of a square matrix with a scalar, the determinant of the matrix will change by the nth power of that scalar.

Step 2: Apply Property

Now, apply the property to find |2A|. As given \(|A|=5\), and the order of A is 3, which is \(n=3\). Also, our scalar (k) in this case is 2. So, |2A| = \(2^3 |A|\).

Step 3: Calculate |2A|

Finally, compute the value of \(2^3 |A|\). So substituting the given |A| value, we get |2A|= \(2^3 \times 5 = 8 \times 5 = 40\).

Key Concepts

Determinant Properties

Understanding the properties of the determinant is crucial when working with matrices. A determinant provides important information about a square matrix, including whether the matrix is invertible and the volume scaling factor for linear transformations associated with the matrix.

One fundamental property is that the determinant of a matrix is a single number that represents a certain product and sum of the matrix's entries. Another key property involves scalar multiplication: when a matrix is multiplied by a scalar, the determinant is multiplied by the scalar raised to the power of the matrix's order. This means that for a 3x3 matrix, if you multiply every entry by 2, the determinant gets multiplied by 2^3 or 8.

Additionally, the determinant of a product of matrices equals the product of their determinants, and the determinant of an identity matrix is always 1. These properties make it easier to compute the determinant in various complex situations.

Matrix Operations

Matrix operations are an essential part of linear algebra and include procedures like addition, subtraction, multiplication, and finding the determinant. In the context of our problem, we focus on how these operations affect the determinant.

To perform scalar multiplication on a matrix, each element of the matrix is multiplied by the scalar. Meanwhile, matrix addition doesn't directly impact the calculation of a determinant but it's still a common matrix operation where corresponding elements of matrices are added together.

Other operations like matrix multiplication can significantly change the determinant. For example, if two matrices are multiplied, the determinant of the resulting matrix is the product of the determinants of the original matrices. Understanding these operations is not only fundamental for solving problems involving determinants but also for broader applications in mathematics and science.

Scalar Multiplication

Scalar multiplication refers to multiplying each entry of a matrix by a constant value, known as a scalar. This operation is one of the simplest yet significant actions we can perform on matrices.

In the given exercise, we deal with scalar multiplication's impact on the determinant. As stated in the property during the solution, when we multiply all entries of a 3x3 matrix, A, by a scalar, k, the determinant of the new matrix is equal to the scalar raised to the power of the matrix's order (which is n=3 for a 3x3 matrix) times the original determinant of A.

Therefore, if each element of a 3x3 matrix is doubled, the determinant of the new matrix, in layman's terms, gets multiplied by two cubed (or eight), not just by two. This property helps us quickly calculate the determinants of matrices after scalar multiplication without having to perform the operation on the matrix and then calculating the determinant from scratch.