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Chapter 3: Problem 38

Let \(A=\left[\begin{array}{ll}{a} & {b} \\ {c} & {d}\end{array}\right]\) and let \(k\) be a scalar. Find a formula that relates det \(k A\) to \(k\) and det \(A\)

Short Answer

Expert verified
\( \text{det}(kA) = k^2 \cdot \text{det}(A) \).

Step by step solution

01

Recall the formula for the determinant of a 2x2 matrix

The determinant of a 2x2 matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\) is given by \( \text{det}(A) = ad - bc \).
02

Identify the matrix resulting from scalar multiplication

Multiplying the scalar \(k\) by the matrix \(A\) results in the matrix \(kA = \begin{bmatrix} ka & kb \ kc & kd \end{bmatrix}\).
03

Calculate the determinant of the scaled matrix

Apply the determinant formula to the scaled matrix \(kA\), yielding \( \text{det}(kA) = (ka)(kd) - (kb)(kc) \).
04

Simplify the expression

Simplifying gives \( \text{det}(kA) = k^2(ad - bc) \).
05

Relate the result to the original determinant

Recognize that \( ad - bc = \text{det}(A) \), thus \( \text{det}(kA) = k^2 \cdot \text{det}(A) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

2x2 matrix
A 2x2 matrix is a simple yet powerful mathematical structure. It is often represented as a square grid with two rows and two columns. Each element in a 2x2 matrix is identified by its position within this grid. A general form of a 2x2 matrix looks like this: \[ A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \]where \(a\), \(b\), \(c\), and \(d\) are the elements of the matrix. These elements can be any real or complex numbers.
  • The term '2x2' refers to the number of rows and columns in the matrix, which is crucial when performing operations like addition, subtraction, and multiplication.
  • Understanding the layout is essential for using the matrix in various calculations, such as determining the determinant or using it in system of equations.
To compute characteristics of a system represented by a matrix, knowing how to work with a 2x2 matrix is often a starting point in learning more complex matrix operations.
scalar multiplication
Scalar multiplication involves multiplying each entry of a matrix by a constant value, known as a scalar. This operation is fundamental in linear algebra and applies to matrices of any size, not just 2x2 matrices.
For matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\) and a scalar \(k\), scalar multiplication yields the new matrix:\[ kA = \begin{bmatrix} ka & kb \ kc & kd \end{bmatrix} \]
  • Each entry in the original matrix is individually multiplied by the scalar. This changes the magnitude of each element but not the ratio between elements.
  • This operation is useful for scaling systems, modifying transformations, or adjusting vectors in spaces without altering their directions.
Scalar multiplication is a direct and straightforward way to adjust the size or intensity of a vector or matrix, as shown through this process of element-wise scaling.
matrix determinant properties
Determinants are important properties of matrices that offer insight into various aspects of linear transformations and systems of equations. For a 2x2 matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is computed as:\[ \text{det}(A) = ad - bc \]This value gives a scalar that represents the matrix's scaling factor, area changes, and whether the matrix can be inverted. Several key properties help in understanding determinants better:
  • The determinant of a product of matrices is the product of their determinants.
  • Scalar multiplication affects the determinant significantly: multiplying a matrix by a scalar \(k\) affects its determinant by \(k^n\), where \(n\) is the order of the matrix (i.e., the number of rows or columns).
  • For a 2x2 matrix, the operation gives \(\text{det}(kA) = k^2 \times \text{det}(A)\), demonstrating how scaling impacts area or volume.
Understanding matrix determinant properties allows us to grasp how transformations affect the spatial configuration in applications like graphics, physics, and more.

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Most popular questions from this chapter

Use the concept of volume to explain why the determinant of a \(3 \times 3\) matrix \(A\) is zero if and only if \(A\) is not invertible. Do not appeal to Theorem 4 in Section \(3.2 .[\text { Hint: Think about }\) the columns of \(A . ]\)

The expansion of a \(3 \times 3\) determinant can be remembered by the following device. Write a second copy of the first two columns to the right of the matrix, and compute the determinant by multiplying entries on six diagonals: Graph cannot copy Add the downward diagonal products and subtract the upward products. Use this method to compute the determinants in Exercises \(15-18 .\) Warning: This trick does not generalize in any reasonable way to \(4 \times 4\) or larger matrices. $$ \left|\begin{array}{rrr}{2} & {-3} & {3} \\ {3} & {2} & {2} \\ {1} & {3} & {-1}\end{array}\right| $$

Compute the determinants in Exercises \(9-14\) by cofactor expansions. At each step, choose a row or column that involves the least amount of computation. $$ \left|\begin{array}{rrrrr}{4} & {0} & {-7} & {3} & {-5} \\ {0} & {0} & {2} & {0} & {0} \\ {7} & {3} & {-6} & {4} & {-8} \\ {5} & {0} & {5} & {2} & {-3} \\\ {0} & {0} & {9} & {-1} & {2}\end{array}\right| $$

Compute the determinants in Exercises \(1-8\) using a cofactor expansion across the first row. In Exercises \(1-4,\) also compute the determinant by a cofactor expansion down the second column. $$ \left|\begin{array}{rrr}{2} & {3} & {-3} \\ {4} & {0} & {3} \\ {6} & {1} & {5}\end{array}\right| $$

In Exercises \(19-24,\) explore the effect of an elementary row operation on the determinant of a matrix. In each case, state the row operation and describe how it affects the determinant. $$ \left[\begin{array}{ll}{3} & {2} \\ {5} & {4}\end{array}\right],\left[\begin{array}{cc}{3} & {2} \\ {5+3 k} & {4+2 k}\end{array}\right] $$
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