Problem 53 - Precalculus Mathematics for Calculus

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

Matrix \( B_1 \) has four square roots by alternating \( a, d \), matrix \( B_2 \) has complex solutions.

Step 1: Understand the Problem

We need to find a matrix \( A \) such that when squared, it results in the given matrix \( B \). We are given two matrices \( B \): \[B_1 = \begin{bmatrix} 4 & 0 \ 0 & 9 \end{bmatrix}\] and \[B_2 = \begin{bmatrix} 1 & 5 \ 0 & 9 \end{bmatrix}.\]

Step 2: Consider the Structure for Matrix \(A\)

Assume the matrix \( A \) is \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \). For each given \( B \), we need to find \( a, b, c, \) and \( d \) such that \( A^2 = B \).

Step 3: Solve for \(A\) in Matrix \(B_1\)

For \( B_1 = \begin{bmatrix} 4 & 0 \ 0 & 9 \end{bmatrix} \), we need \( A^2 = B_1 \). Calculating \( A^2 \) gives:\[A^2 = \begin{bmatrix} a & b \ c & d \end{bmatrix} \cdot \begin{bmatrix} a & b \ c & d \end{bmatrix} = \begin{bmatrix} a^2 + bc & ab + bd \ ac + cd & bc + d^2 \end{bmatrix}. \]We match this with \( B_1 \), resulting in equations \( a^2 + bc = 4 \), \( ab + bd = 0 \), \( ac + cd = 0 \), and \( bc + d^2 = 9 \). Set \( b = c = 0 \) for simplicity, resulting in \( a^2 = 4 \), \( d^2 = 9 \). Hence, \( a = \pm 2 \) and \( d = \pm 3 \). Possible \( A \) are \( \begin{bmatrix} 2 & 0 \ 0 & 3 \end{bmatrix} \), \( \begin{bmatrix} -2 & 0 \ 0 & 3 \end{bmatrix} \), \( \begin{bmatrix} 2 & 0 \ 0 & -3 \end{bmatrix} \), and \( \begin{bmatrix} -2 & 0 \ 0 & -3 \end{bmatrix} \).

Step 4: Solve for \(A\) in Matrix \(B_2\)

For \( B_2 = \begin{bmatrix} 1 & 5 \ 0 & 9 \end{bmatrix} \), we calculate\[ A^2 = \begin{bmatrix} a & b \ c & d \end{bmatrix}^2 = \begin{bmatrix} a^2 + bc & ab + bd \ ac + cd & bc + d^2 \end{bmatrix} = \begin{bmatrix} 1 & 5 \ 0 & 9 \end{bmatrix}. \]The equations are: \( a^2 + bc = 1 \), \( ab + bd = 5 \), \( ac + cd = 0 \), \( bc + d^2 = 9 \). Solving these is more complex without constraints. If \( c = 0 \), then \( ab = 5 \cdot a/d \) which is solvable for specific \( a, d \). Set \( b = 0 \), \( d = 3 \) and \( a = \pm 1 \), which satisfies the simple root assumption. However, more complex roots require further solutions with non-zero \( b \) or \( c \). Examples: Solution is a set of \( ab = 15 \, (Complex Solution) \).

Step 5: Conclusion

There are multiple square roots for the diagonal matrix \( B_1 \). For the second matrix \( B_2 \), simple solutions exist with conditions (like \( c = 0 \)), but generally require more complex analysis.

Key Concepts

Matrix Multiplication

Matrix multiplication is the process of taking two matrices and producing a third matrix by combining the elements according to a specific set of rules. This operation is fundamental in many areas of mathematics and is especially crucial when working with square matrices like the ones involved in finding matrix square roots.

• Row by Column Multiplication: To multiply two matrices, the number of columns in the first matrix must match the number of rows in the second matrix. Each element of the resulting matrix is calculated by multiplying corresponding elements and then summing them up.
• Order Matters: Matrix multiplication is not commutative, meaning that if you multiply matrix A by matrix B, it's not the same as multiplying matrix B by matrix A. The order in which you multiply matrices impacts the result.
• Identity Matrix: In matrix multiplication, an identity matrix acts like the number 1 in regular multiplication. When you multiply any matrix by the identity matrix, you get the original matrix itself.

Understanding how to perform and utilize matrix multiplication is key when solving for matrix square roots, as it's the operation used to verify if a proposed matrix is indeed a square root of another given matrix.

Matrix Algebra

Matrix algebra involves a variety of operations and transformations that can be carried out on matrices. It's a branch of linear algebra that is essential in solving system equations, computer graphics, and optimization problems.

• Addition and Subtraction: These operations are performed by adding or subtracting corresponding elements of matrices with the same dimensions. It’s crucial to always have matrices of the same size to perform these operations.
• Scalar Multiplication: Each element of a matrix can be multiplied by a scalar (a single number), akin to scaling the matrix elements uniformly.
• Transposition: A matrix's transpose is formed by swapping its rows and columns. The transpose is used in various mathematical processes, including finding determinants and inverses.
• Inversion: Some matrices have an inverse, akin to dividing in regular arithmetic. If matrix A has an inverse, then \(A^{-1}\) is such that \(A \cdot A^{-1}\) = Identity matrix. However, not all matrices are invertible.

When finding a square root of a matrix, understanding how these basic operations function will greatly aid your manipulations and solutions. Matrix algebra empowers you to reshape and solve equations that contain matrices.

Diagonal Matrices

Diagonal matrices hold a special place in the study of matrices and linear algebra. A diagonal matrix is one where all the elements outside the main diagonal are zero.

• Simplicity in Multiplication: Because of their structure, multiplying diagonal matrices together, or with other matrices, is particularly straightforward. The product of two diagonal matrices results in another diagonal matrix, with each diagonal element being the product of the corresponding elements.
• Square Roots and Diagonal Matrices: Finding the square root of a diagonal matrix is simpler than most other matrices. For each diagonal entry \(a_{ii}\) of a diagonal matrix \(D\), the square root \(R\) will have diagonal entries that are square roots of the original entries; \(R_{ii} = \sqrt{D_{ii}}\).
• Eigenvalues: The eigenvalues of a diagonal matrix are the values found along its diagonal. This property simplifies many calculations and operations involving these matrices.

Diagonal matrices' straightforward properties make them easier to handle and an excellent example for illustrating concepts like matrix multiplication and finding matrix square roots.