Math Problem Statement

Find a non-zero 2x2 matrix B such that AB = C, where A = [[-8, -1], [24, 3]] and C = [[0, 0], [0, 0]].

Solution

To solve the problem, we need to find a non-zero 2×22 \times 2 matrix BB such that AB=CAB = C, where: A=(81243)A = \begin{pmatrix} -8 & -1 \\ 24 & 3 \end{pmatrix} C=(0000)C = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}

Let B=(abcd)B = \begin{pmatrix} a & b \\ c & d \end{pmatrix}.

We need to perform the matrix multiplication ABAB and set it equal to CC.

  1. Multiply AA by BB: AB=(81243)(abcd)=(8ac8bd24a+3c24b+3d)AB = \begin{pmatrix} -8 & -1 \\ 24 & 3 \end{pmatrix} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} -8a - c & -8b - d \\ 24a + 3c & 24b + 3d \end{pmatrix}

  2. Set ABAB equal to CC: (8ac8bd24a+3c24b+3d)=(0000)\begin{pmatrix} -8a - c & -8b - d \\ 24a + 3c & 24b + 3d \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}

This gives us a system of linear equations:

-8a - c = 0 \\ -8b - d = 0 \\ 24a + 3c = 0 \\ 24b + 3d = 0 \end{cases}$$ 3. Solve the system of equations: From the first equation: $$-8a - c = 0 \implies c = -8a$$ From the second equation: $$-8b - d = 0 \implies d = -8b$$ Substitute $$c$$ in the third equation: $$24a + 3(-8a) = 0 \implies 24a - 24a = 0 \implies 0 = 0$$ This is always true and does not provide new information. Substitute $$d$$ in the fourth equation: $$24b + 3(-8b) = 0 \implies 24b - 24b = 0 \implies 0 = 0$$ This is also always true and does not provide new information. Therefore, we have: $$c = -8a$$ $$d = -8b$$ So, $$B$$ can be any non-zero matrix of the form: $$B = \begin{pmatrix} a & b \\ -8a & -8b \end{pmatrix}$$ For example, we can choose: $$a = 1$$ $$b = 1$$ Then: $$c = -8 \times 1 = -8$$ $$d = -8 \times 1 = -8$$ Thus, one possible solution for $$B$$ is: $$B = \begin{pmatrix} 1 & 1 \\ -8 & -8 \end{pmatrix}$$ Would you like more details or have any questions? Here are five related questions to expand your understanding: 1. How do you perform matrix multiplication? 2. What are the conditions for matrix multiplication to be defined? 3. How can you verify if a given matrix $$B$$ satisfies $$AB = C$$? 4. What are some properties of the zero matrix $$C$$ in matrix equations? 5. How would the solution change if $$C$$ were a non-zero matrix? **Tip:** When solving systems of linear equations derived from matrix equations, always check if the equations provide independent constraints to avoid redundant information.

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Math Problem Analysis

Mathematical Concepts

Matrix Multiplication
Systems of Linear Equations

Formulas

Matrix multiplication formula: (AB)_{ij} = Σ(A_{ik} * B_{kj})
System of equations derived from matrix equality

Theorems

Properties of Matrix Multiplication
Zero Matrix Properties

Suitable Grade Level

Undergraduate