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Math  /  Algebra

Questionrecelculus Q-s Propertes of matrices w. Ras bave gris
If E=[−31−1−331],F=[514−1−3−3]E=\left[\begin{array}{ccc}-3 & 1 & -1 \\ -3 & 3 & 1\end{array}\right], F=\left[\begin{array}{cc}5 & 1 \\ 4 & -1 \\ -3 & -3\end{array}\right], and G=[−5−1−21]G=\left[\begin{array}{cc}-5 & -1 \\ -2 & 1\end{array}\right], is the following statement true or false? E(FG)=(G)FE(F G)=(G) F true false Subm:

Studdy Solution

STEP 1

What is this asking? We need to check if multiplying matrices EE, FF, and GG in two different orders, E(FG)E(FG) and (EF)G(EF)G, gives the same result.
This is about the *associative property* of matrix multiplication. Watch out! Matrix multiplication is *not* always commutative (order matters!), but it *is* associative, meaning we can group the multiplications differently without changing the result.
However, the *dimensions* of the matrices must be compatible for multiplication to even be possible!

STEP 2

1. Calculate FGFG
2. Calculate E(FG)E(FG)
3. Calculate EFEF
4. Calculate (EF)G(EF)G
5. Compare the results

STEP 3

FF is a 3×23 \times 2 matrix and GG is a 2×22 \times 2 matrix.
Since the inner dimensions match (22 and 22), we *can* multiply them, and the result will be a 3×23 \times 2 matrix.

STEP 4

FG=[514−1−3−3][−5−1−21]=[(5)(−5)+(1)(−2)(5)(−1)+(1)(1)(4)(−5)+(−1)(−2)(4)(−1)+(−1)(1)(−3)(−5)+(−3)(−2)(−3)(−1)+(−3)(1)]=[−27−4−18−5210] FG = \begin{bmatrix} 5 & 1 \\ 4 & -1 \\ -3 & -3 \end{bmatrix} \begin{bmatrix} -5 & -1 \\ -2 & 1 \end{bmatrix} = \begin{bmatrix} (5)(-5) + (1)(-2) & (5)(-1) + (1)(1) \\ (4)(-5) + (-1)(-2) & (4)(-1) + (-1)(1) \\ (-3)(-5) + (-3)(-2) & (-3)(-1) + (-3)(1) \end{bmatrix} = \begin{bmatrix} -27 & -4 \\ -18 & -5 \\ 21 & 0 \end{bmatrix}

STEP 5

EE is a 2×32 \times 3 matrix, and FGFG is a 3×23 \times 2 matrix.
The inner dimensions match (33 and 33), so we can multiply them.
The result will be a 2×22 \times 2 matrix.

STEP 6

E(FG)=[−31−1−331][−27−4−18−5210]=[42754−3]E(FG) = \begin{bmatrix} -3 & 1 & -1 \\ -3 & 3 & 1 \end{bmatrix} \begin{bmatrix} -27 & -4 \\ -18 & -5 \\ 21 & 0 \end{bmatrix} = \begin{bmatrix} 42 & 7 \\ 54 & -3 \end{bmatrix}

STEP 7

EE is a 2×32 \times 3 matrix and FF is a 3×23 \times 2 matrix.
The inner dimensions match (33 and 33), so we can multiply them.
The result will be a 2×22 \times 2 matrix.

STEP 8

EF=[−31−1−331][514−1−3−3]=[(−3)(5)+(1)(4)+(−1)(−3)(−3)(1)+(1)(−1)+(−1)(−3)(−3)(5)+(3)(4)+(1)(−3)(−3)(1)+(3)(−1)+(1)(−3)]=[−8−1−6−9]EF = \begin{bmatrix} -3 & 1 & -1 \\ -3 & 3 & 1 \end{bmatrix} \begin{bmatrix} 5 & 1 \\ 4 & -1 \\ -3 & -3 \end{bmatrix} = \begin{bmatrix} (-3)(5) + (1)(4) + (-1)(-3) & (-3)(1) + (1)(-1) + (-1)(-3) \\ (-3)(5) + (3)(4) + (1)(-3) & (-3)(1) + (3)(-1) + (1)(-3) \end{bmatrix} = \begin{bmatrix} -8 & -1 \\ -6 & -9 \end{bmatrix}

STEP 9

EFEF is a 2×22 \times 2 matrix, and GG is a 2×22 \times 2 matrix.
The inner dimensions match (22 and 22), so we can multiply them.
The result will be a 2×22 \times 2 matrix.

STEP 10

(EF)G=[−8−1−6−9][−5−1−21]=[42748−3](EF)G = \begin{bmatrix} -8 & -1 \\ -6 & -9 \end{bmatrix} \begin{bmatrix} -5 & -1 \\ -2 & 1 \end{bmatrix} = \begin{bmatrix} 42 & 7 \\ 48 & -3 \end{bmatrix}

STEP 11

False.

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