Math Problem Statement

Diketahui matriks P = ( 2 1; -3 4 ), Q = ( 6 -8; 2 3 ), R = ( -1 -4; 5 4 ) dan PT adalah transpose dari matriks P, maka nilai dari 3PT - Q + 2R = ...

Solution

The given problem asks for the calculation of the expression:

3PTQ+2R3P^T - Q + 2R

where:

  • P=(2134)P = \begin{pmatrix} 2 & 1 \\ -3 & 4 \end{pmatrix}
  • Q=(6823)Q = \begin{pmatrix} 6 & -8 \\ 2 & 3 \end{pmatrix}
  • R=(1454)R = \begin{pmatrix} -1 & -4 \\ 5 & 4 \end{pmatrix}
  • PTP^T is the transpose of matrix PP.

Steps to solve:

  1. Transpose of matrix PP: PT=(2314)P^T = \begin{pmatrix} 2 & -3 \\ 1 & 4 \end{pmatrix}

  2. Multiply PTP^T by 3: 3PT=3×(2314)=(69312)3P^T = 3 \times \begin{pmatrix} 2 & -3 \\ 1 & 4 \end{pmatrix} = \begin{pmatrix} 6 & -9 \\ 3 & 12 \end{pmatrix}

  3. Calculate 3PTQ3P^T - Q: (69312)(6823)=(0119)\begin{pmatrix} 6 & -9 \\ 3 & 12 \end{pmatrix} - \begin{pmatrix} 6 & -8 \\ 2 & 3 \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 1 & 9 \end{pmatrix}

  4. Multiply matrix RR by 2: 2R=2×(1454)=(28108)2R = 2 \times \begin{pmatrix} -1 & -4 \\ 5 & 4 \end{pmatrix} = \begin{pmatrix} -2 & -8 \\ 10 & 8 \end{pmatrix}

  5. Calculate the final result 3PTQ+2R3P^T - Q + 2R: (0119)+(28108)=(291117)\begin{pmatrix} 0 & -1 \\ 1 & 9 \end{pmatrix} + \begin{pmatrix} -2 & -8 \\ 10 & 8 \end{pmatrix} = \begin{pmatrix} -2 & -9 \\ 11 & 17 \end{pmatrix}

Thus, the final result is:

(291117)\boxed{\begin{pmatrix} -2 & -9 \\ 11 & 17 \end{pmatrix}}

Let me know if you need more details or have further questions.

Related Questions:

  1. What is the formula for matrix transposition?
  2. How do you perform matrix multiplication by a scalar?
  3. Can you explain matrix addition and subtraction?
  4. What are some properties of matrix transposition?
  5. How does matrix multiplication differ from element-wise multiplication?

Tip:

Always ensure matrix dimensions match when performing addition and subtraction, as these operations are only defined for matrices of the same dimensions.

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Math Problem Analysis

Mathematical Concepts

Matrix Transposition
Matrix Scalar Multiplication
Matrix Addition
Matrix Subtraction

Formulas

Transpose of a matrix: (P^T)_ij = P_ji
Matrix scalar multiplication: c * A = c * A_ij
Matrix addition and subtraction: A ± B = (A_ij ± B_ij)

Theorems

Basic Matrix Operations

Suitable Grade Level

Grades 10-12