Question

1. in matrix multiplication $ab \

eq ba$ because commutative property does not generally apply. please show if distributive property works for matrices, such that $a(b+c)= ab +ac$? remember to show step-by-step work and answer the question in complete sentences. (referenced from section 4.3/ 4 points)
$a = \

$$\begin{bmatrix} 0 & 12 \\\\ -7 & 3 \\end{bmatrix}$$

$
$b = \

$$\begin{bmatrix} 2 & 1 \\\\ 0 & -3 \\end{bmatrix}$$

$
$c = \

$$\begin{bmatrix} 0 & 6 \\\\ -8 & 0 \\end{bmatrix}$$

$

Explanation:

Step1: Calculate B+C

Add corresponding entries:

$$ B+C =
LATEXBLOCK0
=
LATEXBLOCK1
$$

Step2: Calculate A(B+C)

Multiply matrix A with B+C:

$$ A(B+C) =
LATEXBLOCK2

LATEXBLOCK3
=
LATEXBLOCK4
=
LATEXBLOCK5
$$

Step3: Calculate AB

Multiply matrix A with B:

$$ AB =
LATEXBLOCK6

LATEXBLOCK7
=
LATEXBLOCK8
=
LATEXBLOCK9
$$

Step4: Calculate AC

Multiply matrix A with C:

$$ AC =
LATEXBLOCK10

LATEXBLOCK11
=
LATEXBLOCK12
=
LATEXBLOCK13
$$

Step5: Calculate AB+AC

Add corresponding entries of AB and AC:

$$ AB+AC =
LATEXBLOCK14
=
LATEXBLOCK15
$$

Answer:

The result of $A(B+C)$ is identical to the result of $AB+AC$, which confirms that the distributive property $A(B+C)=AB+AC$ holds for these given matrices. The final matrix for both expressions is

$$\begin{bmatrix} -96 & -36 \\ -38 & -58 \end{bmatrix}$$

.