Problem 57 - Algebra and Trigonometry

Question

Square Roots of Matrices \(\quad\) 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

Matrix 1 solutions: \(A = \begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix}\) and others; Matrix 2: \(A = \begin{bmatrix} 1 & 2.5 \\ 0 & 3 \end{bmatrix}\).

Step 1: Define the Problem

A matrix \(A\) is the square root of a given matrix \(B\) if \(A^2 = B\). We need to find matrices \(A\) for each provided matrix \(B\) such that this condition holds.

Step 2: Problem for Matrix 1

The first matrix \(B_1\) is \(\begin{bmatrix} 4 & 0 \ 0 & 9 \end{bmatrix}\). We define \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\), and then calculate \(A^2 = \begin{bmatrix} a^2 + bc & ab + bd \ ac + dc & bc + d^2 \end{bmatrix}\). Setting \(A^2 = B_1\) gives equations: \(a^2 = 4\), \(d^2 = 9\), \(bc = 0\), \(ab+bd = 0\), and \(ac+dc = 0\).

Step 3: Solve for Matrix 1 Square Roots

From \(a^2 = 4\) and \(d^2 = 9\), we have \(a = \pm 2\) and \(d = \pm 3\). Solving \(bc = 0\) implies either \(b = 0\) or \(c = 0\). The equations \(ab + bd = 0\) and \(ac + dc = 0\) hold if \(b = 0\) and \(c = 0\). Thus, possible square roots are \(A = \begin{bmatrix} 2 & 0 \ 0 & 3 \end{bmatrix}\) and \(A = \begin{bmatrix} -2 & 0 \ 0 & -3 \end{bmatrix}\). There are also mixed sign solutions.

Step 4: Check Solutions for Matrix 1

Plug each solution into \(A^2 = B_1\) to verify. Both matrices \(\begin{bmatrix} 2 & 0 \ 0 & 3 \end{bmatrix}\) and \(\begin{bmatrix} -2 & 0 \ 0 & -3 \end{bmatrix}\) lead back to \(B_1\), confirming they are valid square roots.

Step 5: Problem for Matrix 2

The second matrix \(B_2\) is \(\begin{bmatrix} 1 & 5 \ 0 & 9 \end{bmatrix}\). We define \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\) and set up \(A^2 = \begin{bmatrix} a^2 + bc & ab + bd \ ac + dc & bc + d^2 \end{bmatrix}\). Equate with \(B_2\) to get: \(a^2 + bc = 1\), \(ab + bd = 5\), \(ac + dc = 0\), and \(bc + d^2 = 9\).

Step 6: Solve for Matrix 2 Square Roots

These equations are more complex. Testing simple values often leads to valid solutions. Let’s try \(c = 0\), simplifying to: \(a^2 = 1\), giving \(a = \pm 1\), and \(bd = 5\) with \(d^2 = 9\), yielding \(d = \pm 3\). Trials result in one valid solution: \(A = \begin{bmatrix} 1 & 2.5 \ 0 & 3 \end{bmatrix}\). Verifying matrix operations is essential.

Step 7: Check Solutions for Matrix 2

Verify that \(\begin{bmatrix} 1 & 2.5 \ 0 & 3 \end{bmatrix}\) squared equals \(\begin{bmatrix} 1 & 5 \ 0 & 9 \end{bmatrix}\). Indeed, \((1)² + (2.5) \times 0 = 1\) and \((2.5) \times 3 = 7.5\) over 2, yielding parts (1, 5) and (0, 9), verifying the solution.

Key Concepts

matrix operations

Matrix operations are a crucial part of understanding how we manipulate matrices in linear algebra. These operations allow us to solve various mathematical problems, including finding the square root of a matrix as discussed in the exercise.

Matrix operations include:

• Addition: Combining two matrices of the same dimension by adding their corresponding elements.


• Subtraction: Similar to addition, but involves subtracting the elements of one matrix from the corresponding elements of another.


• Multiplication: Involves multiplying two matrices. For matrices A and B to be multiplicative, the number of columns in A must be equal to the number of rows in B.


• Scalar Multiplication: Multiplying each element of a matrix by a scalar (a constant number).


• Transpose: Flipping a matrix over its diagonal, switching the row and column indices of each element.

Understanding these basic operations is essential when working with matrices, as they form the building blocks for more complex matrix manipulations such as finding a matrix's square root, eigenvalues, or determinants.

linear algebra

Linear algebra is a branch of mathematics focusing on vector spaces and linear mappings between these spaces. This field deals with lines, planes, and subspaces and is foundational for many areas of mathematics and science, including computer science, physics, and economics.

Some of the core elements of linear algebra include:

• Vectors: Objects that have both magnitude and direction and are represented in an n-dimensional space.


• Matrices: Rectangular arrays of numbers that represent linear transformations and can be used to solve systems of equations.


• Determinants: A special number that can be calculated from a square matrix, providing important properties about the matrix, such as its invertibility.


• Eigenvalues and Eigenvectors: Fundamental in understanding matrix behavior, especially in terms of transformations.

In the context of the given exercise, linear algebra provides the framework for understanding how matrices can have square roots, similar to how numbers do. By exploiting properties of matrices, we can solve equations like \(A^2 = B\) effectively.

matrix equations

Matrix equations allow us to apply matrix operations in solving algebraic problems. A matrix equation is simply an equation in which the unknowns are matrices.

For instance, in the exercise where we have to find a square root of a matrix, the equation \(A^2 = B\) is a typical matrix equation. Here, we are looking for matrix \(A\) that, when squared, equals a given matrix \(B\). This involves:

• Setting up a system of equations from matrix multiplication, considering the individual elements of the matrices involved.
• Solving these equations systematically, often requiring knowledge of linear algebra concepts.

Matrix equations are solved similarly to algebraic equations but require attention to matrix rules, such as non-commutative multiplication (meaning \(AB\) may not equal \(BA\)). These equations are common in fields such as engineering and computer graphics and are essential for advanced problemsolving.