Problem 57 - Precalculus: Mathematical for Calculus

Question

\(A\) square root of a matrix \(B\) is a matrix \(A\) with the property that \(A^{2}=B\). (This is the same definition as for a square root of a number.) Find as many square roots as you can of each matrix: $$\left[\begin{array}{ll} 4 & 0 \\ 0 & 9 \end{array}\right] \quad\left[\begin{array}{ll} 1 & 5 \\ 0 & 9 \end{array}\right]$$ [Hint: If \(A=$$\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]\) write the equations that \(a, b, c,\) and \(d\) would have to satisfy if \(A\) is the square root of the given matrix.]

Step-by-Step Solution

Answer

The square roots are \(\left[\begin{array}{ll}2 & 0 \\ 0 & 3\end{array}\right]\), \(\left[\begin{array}{ll}-2 & 0 \\ 0 & 3\end{array}\right]\), \(\left[\begin{array}{ll}2 & 0 \\ 0 & -3\end{array}\right]\), \(\left[\begin{array}{ll}-2 & 0 \\ 0 & -3\end{array}\right]\) for the first matrix and \(\left[\begin{array}{ll}1 & 5/3 \\ 0 & 3\end{array}\right]\) for the second matrix.

Step 1: Define the Square Root Matrix

Let's consider a matrix \( A \) in the form \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \). We will calculate \( A^2 \) to compare it with the given matrices.

Step 2: Calculate \( A^2 \)

Compute the square of matrix \( A \):\[A^2 = \begin{pmatrix} a & b \ c & d \end{pmatrix} \begin{pmatrix} a & b \ c & d \end{pmatrix} = \begin{pmatrix} a^2 + bc & ab + bd \ ac + cd & bc + d^2 \end{pmatrix}.\]

Step 3: Find Square Root for First Matrix

Set \( A^2 \) equal to the first matrix \( \begin{pmatrix} 4 & 0 \ 0 & 9 \end{pmatrix} \). This provides the following equations:- \( a^2 + bc = 4 \)- \( ab + bd = 0 \)- \( ac + cd = 0 \)- \( bc + d^2 = 9 \).Solving these, it's clear \( b = c = 0 \), and thus \( a^2 = 4 \), \( d^2 = 9 \). Therefore, possible solutions are \( a = \pm 2 \) and \( d = \pm 3 \). The possible square roots are:- \( \begin{pmatrix} 2 & 0 \ 0 & 3 \end{pmatrix} \)- \( \begin{pmatrix} -2 & 0 \ 0 & 3 \end{pmatrix} \)- \( \begin{pmatrix} 2 & 0 \ 0 & -3 \end{pmatrix} \)- \( \begin{pmatrix} -2 & 0 \ 0 & -3 \end{pmatrix} \).

Step 4: Find Square Root for Second Matrix

Set \( A^2 \) equal to the second matrix \( \begin{pmatrix} 1 & 5 \ 0 & 9 \end{pmatrix} \). This provides the following equations:- \( a^2 + bc = 1 \)- \( ab + bd = 5 \)- \( ac + cd = 0 \)- \( bc + d^2 = 9 \).Assume \( c = 0 \) (since \( ac + cd = 0 \)), then \( d^2 = 9 \) implies \( d = 3 \) or \( d = -3 \). Solving for \( a \) and \( b \):- If \( d = 3 \), then \( a^2 = 1 \) gives \( a = 1 \), \( b = 5/3 \), leading to the solution \( \begin{pmatrix} 1 & 5/3 \ 0 & 3 \end{pmatrix} \).- If \( d = -3 \), then it's not possible since \( ab + bd \) can't equal 5 with integer \( a \) and \( b \).Thus, only one valid square root matrix is \( \begin{pmatrix} 1 & 5/3 \ 0 & 3 \end{pmatrix} \).

Key Concepts

Matrix Algebra

Matrix algebra forms the basis for many mathematical computations and applications in various fields such as physics and engineering. At its core, matrix algebra involves the operations and manipulations of matrices, which are rectangular arrays of numbers. These operations include matrix addition, subtraction, multiplication, and finding inverses.

When we talk about the square root of a matrix, we mean finding another matrix that, when multiplied by itself, yields the original matrix. This is analogous to finding the square root of a number, where you are finding a number which, when squared, gives back the original number. However, dealing with matrices is more complex due to the multi-dimensional nature of these arrays.

Not every matrix has a square root, and some matrices can have more than one square root, especially when they are diagonalizable. Matrices can be represented by various forms, and understanding matrix algebra helps in determining such transformations and solutions to equations involving matrices.

Square Matrices

Square matrices are a special type of matrix in which the number of rows and columns are equal. These matrices often arise in mathematical applications and represent linear transformations that map vector spaces onto themselves.

A key characteristic of square matrices is that they can be used to represent systems of linear equations. Additionally, they have properties such as determinants and traces, which can provide information about the matrix itself, including whether it is invertible or singular.

The concept of a matrix square root is most commonly applied to square matrices. For instance, finding the square root of a diagonal matrix is straightforward when considering each diagonal element’s square root. For the matrix \( \begin{pmatrix} 4 & 0 \ 0 & 9 \end{pmatrix} \), we see that the square roots could be \( \begin{pmatrix} \pm 2 & 0 \ 0 & \pm 3 \end{pmatrix} \). The simplicity of square matrices provides an easier avenue to find these roots due to their orderly structure.

Matrix Operations

Matrix operations are fundamental processes in linear algebra, crucial for solving equations and transformations. Key operations include addition, subtraction, and multiplication of matrices, as well as more advanced operations like finding the determinant and the inverse.

Multiplying matrices requires the summation of products of their rows and columns, making it computationally intensive. However, when dealing with square root computations, the operation focuses on finding matrices that fulfill specific equations, such as \( A^2 = B \) for a matrix \( B \), as described in the given problem.

For instance, in solving for the square root of \( \begin{pmatrix} 1 & 5 \ 0 & 9 \end{pmatrix} \), the operations required combine logical deductions and algebraic manipulations: setting up equations, assuming potential values for variables, and verifying whether they satisfy all the conditions. Each step involves matrix multiplication, enabling us to check if the calculated matrix returned the original matrix upon squaring, thereby confirming it as a square root.

Understanding matrix operations is critical when solving complex problems involving matrices, as these operations form the underlying steps to reach a solution.