Math Problem Statement
Given a 2x2 matrix multiplied by X to give a 2x1 matrix, find the order of matrix X and solve for X.
Solution
Let's break down the given problem:
You are provided with the following matrix equation:
2 & 1 \\ -3 & 4 \end{bmatrix} \mathbf{X} = \begin{bmatrix} 7 \\ 6 \end{bmatrix}$$ ### (i) The order of the matrix $$\mathbf{X}$$: - The matrix on the left-hand side is a 2x2 matrix (2 rows and 2 columns). - The matrix on the right-hand side is a 2x1 matrix (2 rows and 1 column). Since the left-hand side is a multiplication of a 2x2 matrix and $$\mathbf{X}$$, for the multiplication to be valid, $$\mathbf{X}$$ must be a 2x1 matrix. Therefore, the order of the matrix $$\mathbf{X}$$ is **2x1**. ### (ii) The matrix $$\mathbf{X}$$: To find $$\mathbf{X}$$, we need to solve the system of equations that arises from the matrix multiplication: $$\begin{bmatrix} 2 & 1 \\ -3 & 4 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 7 \\ 6 \end{bmatrix}$$ This gives us two equations: 1. $$2x_1 + x_2 = 7$$ 2. $$-3x_1 + 4x_2 = 6$$ Let's solve this system. From the first equation, solve for $$x_2$$: $$x_2 = 7 - 2x_1$$ Substitute this into the second equation: $$-3x_1 + 4(7 - 2x_1) = 6$$ Simplifying: $$-3x_1 + 28 - 8x_1 = 6$$ $$-11x_1 = -22$$ $$x_1 = 2$$ Now, substitute $$x_1 = 2$$ back into the equation $$x_2 = 7 - 2x_1$$: $$x_2 = 7 - 2(2) = 3$$ Thus, the matrix $$\mathbf{X}$$ is: $$\mathbf{X} = \begin{bmatrix} 2 \\ 3 \end{bmatrix}$$ --- Would you like further details or have any questions? Here are some additional questions to expand your understanding: 1. How do you check the validity of matrix multiplication? 2. Can the order of a matrix affect its ability to be multiplied by another matrix? 3. How would you solve a system of equations using matrix inverses? 4. What would change if the matrix on the right-hand side was different? 5. How does the determinant of the left-hand matrix influence the solvability of the system? **Tip:** When solving systems of equations using matrices, always check the determinant of the coefficient matrix. If it’s zero, the system may have no unique solution.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix multiplication
System of equations
Algebra
Formulas
Matrix multiplication formula: AB = C
Equation 1: 2x1 + x2 = 7
Equation 2: -3x1 + 4x2 = 6
Theorems
Matrix multiplication rules
System of linear equations
Suitable Grade Level
Grades 9-12