Math Problem Statement

Given matrices A = [[x, 1], [y, 2]] and B = [[x, x/2], [y, 2]] such that AB is a null matrix. Find: (a) the order of the null matrix, (b) possible values of x and y.

Solution

The problem visible in the image reads:

86.
Given, matrix A=[x1y2]A = \begin{bmatrix} x & 1 \\ y & 2 \end{bmatrix} and B=[xx2y2]B = \begin{bmatrix} x & \frac{x}{2} \\ y & 2 \end{bmatrix} such that ABAB is a null matrix. Find:

  • (a) order of the null matrix.
  • (b) possible values of xx and yy.

Let me break this down.

Part (a): Order of the Null Matrix

To find the order of the null matrix, we first need to calculate the product ABAB. Since AA is a 2×22 \times 2 matrix and BB is also a 2×22 \times 2 matrix, their product will result in another 2×22 \times 2 matrix. Hence, the order of the null matrix is 2×22 \times 2.

Part (b): Finding Possible Values of xx and yy

We calculate the matrix multiplication ABAB and equate it to the null matrix.

  1. Matrix A:
    A=[x1y2]A = \begin{bmatrix} x & 1 \\ y & 2 \end{bmatrix}

  2. Matrix B:
    B=[xx2y2]B = \begin{bmatrix} x & \frac{x}{2} \\ y & 2 \end{bmatrix}

Now, multiplying AA and BB:

= \begin{bmatrix} (x \cdot x + 1 \cdot y) & (x \cdot \frac{x}{2} + 1 \cdot 2) \\ (y \cdot x + 2 \cdot y) & (y \cdot \frac{x}{2} + 2 \cdot 2) \end{bmatrix}$$ Simplifying each element of the resulting matrix: - Top-left: $$x^2 + y$$ - Top-right: $$\frac{x^2}{2} + 2$$ - Bottom-left: $$xy + 2y$$ - Bottom-right: $$\frac{xy}{2} + 4$$ Thus: $$AB = \begin{bmatrix} x^2 + y & \frac{x^2}{2} + 2 \\ xy + 2y & \frac{xy}{2} + 4 \end{bmatrix}$$ For $$AB$$ to be a null matrix, all entries must be zero: 1. $$x^2 + y = 0$$ (Equation 1) 2. $$\frac{x^2}{2} + 2 = 0$$ (Equation 2) 3. $$xy + 2y = 0$$ (Equation 3) 4. $$\frac{xy}{2} + 4 = 0$$ (Equation 4) ### Solving the system of equations: From **Equation 2**: $$\frac{x^2}{2} + 2 = 0 \implies x^2 = -4 \implies \text{No real solutions for } x.$$ Thus, the system has no real solution. Therefore, there are no real values of $$x$$ and $$y$$ that satisfy the condition for $$AB$$ to be a null matrix. Do you want a deeper explanation, or have any specific questions about the process? ### Related Questions: 1. How do you multiply two matrices in general? 2. What conditions must be met for two matrices to be multiplied? 3. How do you determine the dimensions of a matrix product? 4. Can matrices with complex elements satisfy the given system of equations? 5. How do you interpret a null matrix in the context of matrix operations? **Tip**: When solving systems of equations derived from matrix multiplication, simplify each entry step by step to avoid errors.

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Math Problem Analysis

Mathematical Concepts

Matrix Multiplication
System of Equations

Formulas

Matrix multiplication formula: (A * B)_ij = Σ A_ik * B_kj
Equation of a null matrix: AB = 0

Theorems

Matrix Nullity Theorem

Suitable Grade Level

Grades 10-12