Problem 53 - Algebra and Trigonometry

Question

Square Roots of Matrices 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

Possible square roots are \(\begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix}\), \(\begin{bmatrix} -2 & 0 \\ 0 & 3 \end{bmatrix}\), \(\begin{bmatrix} 2 & 0 \\ 0 & -3 \end{bmatrix}\), \(\begin{bmatrix} -2 & 0 \\ 0 & -3 \end{bmatrix}\) for \(B_1\) and \(\begin{bmatrix} 2 & 1 \\ 0 & 3 \end{bmatrix}\) for \(B_2\).

Step 1: Identify the Matrix Equation

We are given the matrices \(B_1 = \begin{bmatrix} 4 & 0 \ 0 & 9 \end{bmatrix}\) and \(B_2 = \begin{bmatrix} 1 & 5 \ 0 & 9 \end{bmatrix}\). We need to find matrices \(A_1\) and \(A_2\) such that \(A_1^2 = B_1\) and \(A_2^2 = B_2\). Let \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\). The relationship we need to satisfy is \(A^2 = B\).

Step 2: Set Up Equations for Matrix \(B_1\)

We need \(A_1^2 = \begin{bmatrix} 4 & 0 \ 0 & 9 \end{bmatrix}\). Expanding \(A_1^2\) gives \[\begin{bmatrix} a & b \ c & d \end{bmatrix}^2 = \begin{bmatrix} a^2 + bc & ab + bd \ ac + cd & bc + d^2 \end{bmatrix}.\]For \(A_1\), this means the equations are: 1. \(a^2 = 4\)2. \(d^2 = 9\)3. \(bc = 0\)4. \(ab + bd = 0\)5. \(ac + cd = 0\)

Step 3: Solve for Possible Values in \(B_1\)

From \(a^2 = 4\), \(a\) can be \(\pm 2\). From \(d^2 = 9\), \(d\) can be \(\pm 3\). Since \(bc = 0\), either \(b = 0\) or \(c = 0\). Assuming both are zero, we satisfy all other equations since \(ab + bd = 0\) and \(ac + cd = 0\) simplify to zero.Therefore, potential square roots for \(B_1\) could be \(\begin{bmatrix} 2 & 0 \ 0 & 3 \end{bmatrix}\), \(\begin{bmatrix} -2 & 0 \ 0 & 3 \end{bmatrix}\), \(\begin{bmatrix} 2 & 0 \ 0 & -3 \end{bmatrix}\), and \(\begin{bmatrix} -2 & 0 \ 0 & -3 \end{bmatrix}\).

Step 4: Set Up Equations for Matrix \(B_2\)

We need \(A_2^2 = \begin{bmatrix} 1 & 5 \ 0 & 9 \end{bmatrix}\). Expand \(A_2^2\) and set it equal to \(B_2\):1. \(a^2 + bc = 1\)2. \(ab + bd = 5\)3. \(ac + cd = 0\)4. \(d^2 = 9\)

Step 5: Solve the System for \(B_2\)

From \(d^2 = 9\), \(d\) can be \(\pm 3\). Assume \(d = 3\). This gives:1. \(ab + 3b = 5\)2. \(ac + 3c = 0\)We attempt to solve it assuming simple values for you to explore manually this case. Let's consider a particular case: if \(b = 1\),/then \(a + 3 = 5\) so \(a = 2\), and since \(c = -\frac{3c}{a + 3}\), the simplest solution setting \(c = 0\) holds all terms intact.Thus, one possible matrix \(A_2 = \begin{bmatrix} 2 & 1 \ 0 & 3 \end{bmatrix}\). However, other values of \(a, b, \text{or } c\) might yield possible results due to assumption variations.

Key Concepts

Square Roots of Matrices

In linear algebra, the concept of the square root of a matrix parallels the definition of a square root of a number. If we have a matrix \( B \), a matrix \( A \) is considered a square root of \( B \) if \( A^2 = B \). This means that when you multiply the matrix \( A \) by itself, you should end up with the original matrix \( B \).

For example, if you consider the matrix \([4, 0; 0, 9]\), a potential square root is \([2, 0; 0, 3]\). In this case:

• Calculate \( A^2 \) by performing the matrix multiplication of \( A \) by itself.
• Check if the result equals \( B \).

Given \(\begin{bmatrix} 2 & 0 \ 0 & 3 \end{bmatrix}^2 = \begin{bmatrix} 4 & 0 \ 0 & 9 \end{bmatrix}\), confirms that \( \begin{bmatrix} 2 & 0 \ 0 & 3 \end{bmatrix} \) is indeed a square root of matrix \( B \).

Notably, since squaring a negative number yields a positive result, negative matrices like \( \begin{bmatrix} -2 & 0 \ 0 & -3 \end{bmatrix} \) can also be square roots.

Matrix Equations

Matrix equations are fundamental in linear algebra and often involve setting up a system of equations to solve unknown elements of a matrix. For matrix square roots, you need to resolve a set of equations derived from the condition \( A^2 = B \). This requires an understanding of matrix multiplication.

Take, for instance, the matrix \( A \) expressed as \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \). When squaring this matrix to equal another matrix \( B \), we get:

• \( a^2 + bc = \text{some element in B} \)
• \( ab + bd = \text{some element in B} \)
• \( ac + cd = \text{some element in B} \)
• \( d^2 = \text{some element in B} \)

Breaking down these equations helps in identifying possible values for the variables \( a, b, c, \) and \( d \). These equations are critical in ensuring that our guessed or calculated matrix \( A \) conforms precisely to the conditions set by matrix \( B \).

Solving these equations may depend on employing both algebraic manipulation and sometimes intuitive guesswork based on properties the matrix satisfies, such as symmetry or special numbers like zero.

Linear Algebra

Linear algebra is the branch of mathematics concerning linear equations, linear functions, and their representations through matrices and vector spaces. A crucial part of linear algebra is understanding matrices, as they are not only numerical arrays but also function with operations that mirror many algebraic operations.

For example, determining the square root of a matrix is an operation that requires knowledge of:

• Matrix multiplication rules, which dictate the results of multiplying two matrices.
• The symmetry and properties of special matrices like the identity matrix and zero matrix.

Understanding matrices is essential, as they are used in systems of equations that are vital for analyzing linear transformations and solving real-world problems in engineering and computer science.

Exploring the properties of matrices can help to deepen the comprehension of abstract spaces and dimensions in theory, as well as practical systems of linear equations and transformations.

Matrix Operations

Matrix operations form the core of solving and manipulating matrix equations in linear algebra. Some essential matrix operations include addition, subtraction, multiplication, and finding inverses, but particularly here, focusing on multiplication helps balance equations like \( A^2 = B \).

To find the square root of a matrix via operations, one should be familiar with:

• Expanding squared matrices and managing overlapping terms.
• Simplifying results back into a clean, standard matrix form.
• Checking results against identity matrices or other known solutions for confirmation.

It’s like solving a puzzle—you must consider each piece (or number in the matrix) individually and then together as part of the full picture.

A special point of interest can also arise in particular operations: the determinant. But remember, while determinants are essential for more advanced matrix concepts, they don’t directly influence the square root operation. Focus should remain on ensuring calculations remain mathematically sound at each step.