Math Problem Statement

Jawabannya

Solve the matrix equation involving the unknowns a, b, c, and d.

Solution

The equation in the image appears to involve matrix operations. Here's a step-by-step interpretation:

Given equation:

2(a231)+(410b)=(32c4)(2d13)2 \begin{pmatrix} a & 2 \\ -3 & 1 \end{pmatrix} + \begin{pmatrix} 4 & -1 \\ 0 & b \end{pmatrix} = \begin{pmatrix} 3 & 2 \\ c & 4 \end{pmatrix} \begin{pmatrix} 2 & d \\ 1 & 3 \end{pmatrix}

We will solve this equation step by step.

Step 1: Simplify the left-hand side (LHS)

  1. First matrix scalar multiplication: 2(a231)=(2a462)2 \begin{pmatrix} a & 2 \\ -3 & 1 \end{pmatrix} = \begin{pmatrix} 2a & 4 \\ -6 & 2 \end{pmatrix}

  2. Adding the matrices on the LHS: (2a462)+(410b)=(2a+44+(1)6+02+b)\begin{pmatrix} 2a & 4 \\ -6 & 2 \end{pmatrix} + \begin{pmatrix} 4 & -1 \\ 0 & b \end{pmatrix} = \begin{pmatrix} 2a + 4 & 4 + (-1) \\ -6 + 0 & 2 + b \end{pmatrix} So, the left-hand side becomes: (2a+4362+b)\begin{pmatrix} 2a + 4 & 3 \\ -6 & 2 + b \end{pmatrix}

Step 2: Multiply the matrices on the right-hand side (RHS)

Now let's multiply the two matrices on the right-hand side: (32c4)(2d13)\begin{pmatrix} 3 & 2 \\ c & 4 \end{pmatrix} \begin{pmatrix} 2 & d \\ 1 & 3 \end{pmatrix}

Multiplying the matrices:

  • Top left: 3×2+2×1=6+2=83 \times 2 + 2 \times 1 = 6 + 2 = 8
  • Top right: 3×d+2×3=3d+63 \times d + 2 \times 3 = 3d + 6
  • Bottom left: c×2+4×1=2c+4c \times 2 + 4 \times 1 = 2c + 4
  • Bottom right: c×d+4×3=cd+12c \times d + 4 \times 3 = cd + 12

So, the RHS is: (83d+62c+4cd+12)\begin{pmatrix} 8 & 3d + 6 \\ 2c + 4 & cd + 12 \end{pmatrix}

Step 3: Equate both sides

We now equate the corresponding elements from both sides of the equation.

(2a+4362+b)=(83d+62c+4cd+12)\begin{pmatrix} 2a + 4 & 3 \\ -6 & 2 + b \end{pmatrix} = \begin{pmatrix} 8 & 3d + 6 \\ 2c + 4 & cd + 12 \end{pmatrix}

This gives us the following system of equations:

  1. 2a+4=82a + 4 = 8
  2. 3=3d+63 = 3d + 6
  3. 6=2c+4-6 = 2c + 4
  4. 2+b=cd+122 + b = cd + 12

Step 4: Solve the system of equations

  1. 2a+4=82a=4a=22a + 4 = 8 \Rightarrow 2a = 4 \Rightarrow a = 2
  2. 3=3d+63d=36=3d=13 = 3d + 6 \Rightarrow 3d = 3 - 6 = -3 \Rightarrow d = -1
  3. 6=2c+42c=10c=5-6 = 2c + 4 \Rightarrow 2c = -10 \Rightarrow c = -5
  4. 2+b=cd+122+b=(5)(1)+122+b=5+12=17b=152 + b = cd + 12 \Rightarrow 2 + b = (-5)(-1) + 12 \Rightarrow 2 + b = 5 + 12 = 17 \Rightarrow b = 15

Final solution:

a=2,b=15,c=5,d=1a = 2, \, b = 15, \, c = -5, \, d = -1

Let me know if you need further details or explanations!

Here are 5 related questions for deeper exploration:

  1. What properties of matrix multiplication are essential when solving systems like this?
  2. How do scalar multiplication and matrix addition work in matrix algebra?
  3. Could you solve the system using inverse matrices instead of direct element comparison?
  4. How does changing the order of matrix multiplication affect the result?
  5. Can matrix multiplication be performed with non-square matrices?

Tip: When multiplying matrices, remember that the number of columns in the first matrix must equal the number of rows in the second matrix for the multiplication to be valid!

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Math Problem Analysis

Mathematical Concepts

Matrix Algebra
Matrix Multiplication
Systems of Equations

Formulas

Matrix Scalar Multiplication
Matrix Addition
Matrix Multiplication

Theorems

Properties of Matrix Multiplication
Systems of Linear Equations

Suitable Grade Level

Grades 10-12