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Ch. 6 - Matrices and Determinants
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 6 - Matrices and DeterminantsProblema 17
Capitolo 7, Problema 17

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. X - A = B

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Identify the given matrices: \( A = \begin{bmatrix} -3 & -7 \\ 2 & -9 \\ 5 & 0 \end{bmatrix} \) and \( B = \begin{bmatrix} -5 & -1 \\ 0 & 0 \\ 3 & -4 \end{bmatrix} \). The equation to solve is \( X - A = B \).
To isolate \( X \), add matrix \( A \) to both sides of the equation: \( X - A + A = B + A \), which simplifies to \( X = B + A \).
Perform matrix addition by adding corresponding elements of matrices \( B \) and \( A \). For example, the element in the first row and first column of \( X \) is \( B_{11} + A_{11} = -5 + (-3) \).
Continue adding each corresponding element: \( X_{12} = B_{12} + A_{12} = -1 + (-7) \), \( X_{21} = B_{21} + A_{21} = 0 + 2 \), and so on for all elements.
Write the resulting matrix \( X \) after completing the addition of all corresponding elements from \( A \) and \( B \).

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Matrix Addition and Subtraction

Matrix addition and subtraction involve combining corresponding elements from two matrices of the same dimensions. To subtract matrix A from matrix X, each element in A is subtracted from the corresponding element in X. This operation is essential for solving equations like X - A = B.
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Solving Matrix Equations

To solve matrix equations such as X - A = B, isolate the unknown matrix X by performing inverse operations. Here, adding matrix A to both sides yields X = B + A. Understanding how to manipulate matrices algebraically is crucial for finding the solution.
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Matrix Dimensions and Compatibility

Matrix operations require that matrices have compatible dimensions. Both A and B must be the same size to perform addition or subtraction. Recognizing the dimensions ensures valid operations and helps avoid errors when solving matrix equations.
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Introduction to Matrices
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Use the fact that if A=[abcd]A=\(\begin{bmatrix}\)a & b\\ c & d\(\end{bmatrix}\), then A−1=1ad−bc[d−b−ca]A^{-1}=\(\frac{1}{ad-bc}\)\(\begin{bmatrix}\)d & -b\\ -c & a\(\end{bmatrix}\) to find the inverse of each matrix, if possible. Check that AA−1=I2AA^{-1} = I_2 and A−1A=I2A^{-1}A = I_2.

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