Let A=[−3−72−950]A = \begin{bmatrix} -3 & -7 \\ 2 & -9 \\ 5 & 0 \end{bmatrix} and B=[−5−1003−4]B = \begin{bmatrix} -5 & -1 \\ 0 & 0 \\ 3 & -4 \end{bmatrix}. Solve each matrix equation for X. 4A + 3B = - 2X

Ch. 6 - Matrices and Determinants

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Blitzer 8th Edition

Ch. 6 - Matrices and Determinants

문제 25

7장, 문제 25

Let $A = [\begin{matrix} - 3 & - 7 \\ 2 & - 9 \\ 5 & 0 \end{matrix}]$ and $B = [\begin{matrix} - 5 & - 1 \\ 0 & 0 \\ 3 & - 4 \end{matrix}]$ . Solve each matrix equation for X. 4A + 3B = - 2X

검증된 단계별 안내

First, write down the given matrix equation: $4A + 3B = -2X$.

To solve for $X$, isolate it by dividing both sides of the equation by $-2$, which gives: $X = -\frac{1}{2}(4A + 3B)$.

Next, calculate the matrix \$4A$ by multiplying each element of matrix $A$ by 4.

Then, calculate the matrix \$3B$ by multiplying each element of matrix $B$ by 3.

Add the resulting matrices \$4A$ and $3B$ element-wise, then multiply the sum by \(-\frac{1}{2}$ to find matrix \)X$.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Matrix Operations

Matrix operations include addition, subtraction, and scalar multiplication, which are essential for manipulating matrices. Understanding how to multiply a matrix by a scalar and add or subtract matrices element-wise is crucial for solving matrix equations like 4A + 3B = -2X.

Solving Matrix Equations

Solving matrix equations involves isolating the unknown matrix by performing inverse operations. For an equation like 4A + 3B = -2X, you must first combine known matrices and then divide or multiply by the inverse scalar to find X.

Scalar Multiplication and Division of Matrices

Scalar multiplication involves multiplying every element of a matrix by a constant. To solve for X in -2X = 4A + 3B, you multiply matrices A and B by scalars 4 and 3 respectively, then divide the resulting matrix by -2 to isolate X.

Let A=[−3−72−950]A = \(\begin{bmatrix}\) -3 & -7 \\ 2 & -9 \\ 5 & 0 \(\end{bmatrix}\) and B=[−5−1003−4]B = \(\begin{bmatrix}\) -5 & -1 \\ 0 & 0 \\ 3 & -4 \(\end{bmatrix}\). Solve each matrix equation for X. 4A + 3B = - 2X