// Numbas version: finer_feedback_settings {"name": "Scalar Multiplication", "metadata": {"description": "

Exam to test students understanding of matrices multiplied by a scalar.

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Scalar Multiple of a Matrix

\n

Carry out the scalar multiplication on the following matrices

", "advice": "

We carry out the calculation by multiplying each element by the constant:

\n

a)

\n

$$
\\begin{aligned}
\\var{k1}A &= \\var{k1}\\var{A} \\\\
&=\\begin{pmatrix}
\\var{k1} \\times \\var{A[0][0]} & \\var{k1} \\times \\var{A[0][1]}& \\var{k1} \\times \\var{A[0][2]}\\\\
\\var{k1} \\times \\var{A[1][0]} & \\var{k1} \\times \\var{A[1][1]}& \\var{k1} \\times \\var{A[1][2]}\\\\
\\var{k1} \\times \\var{A[2][0]} & \\var{k1} \\times \\var{A[2][1]} & \\var{k1} \\times \\var{A[2][2]}\\\\
\\end{pmatrix} \\\\
&= \\var{ad1} 
\\end{aligned}
$$

\n

b)

\n

Using the same technique, we get:

\n

$$
\\begin{aligned}
\\var{k2}B &= \\var{k2}\\var{B} \\\\
&= \\var{ad2}
\\end{aligned}
$$ 

\n

c)

\n

$$
\\begin{aligned}
\\var{k3}C &= \\var{k3}\\var{C} \\\\
&= \\var{ad3}
\\end{aligned}
$$ 

\n

d)

\n

$$
\\begin{aligned}
\\var{k4}D &= \\var{k4}\\var{D} \\\\
&= \\var{ad4}
\\end{aligned}
$$ 

\n

e)

\n

$$
\\begin{aligned}
\\var{k5}E &= \\var{k5}\\var{EE} \\\\
&= \\var{ad5}
\\end{aligned}
$$

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Let:

\n

$$
A=\\var{A}
$$

\n

Calculate $\\var{k1} A$:

\n

$\\var{k1}A=$ [[0]]

\n

\n

\n

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Let:

\n

$$
B=\\var{B}
$$

\n

Calculate $\\var{k2}B$:

\n

$\\var{k2}B=$ [[0]]

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Let:

\n

$$
C=\\var{C}
$$

\n

Calculate $\\var{k3}C$:

\n

$\\var{k3}C=$ [[0]]

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Let:

\n

$$
D=\\var{D}
$$

\n

Calculate $\\var{k4}D$:

\n

$\\var{k4}D=$ [[0]]

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Let:

\n

$$
E=\\var{EE}
$$

\n

Calculate $\\var{k5}E$:

\n

$\\var{k5}E=$ [[0]]

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Addition and Subraction with Scalar Multiples

\n

For each of the given matrices, complete the calculation using matrix addition, subtraction and scalar multiplication

", "advice": "

We are asked to carry out combinations of matrix operations. We combine the rules that we have used previously to determine our answer:

\n

a)

\n

If $A=\\var{A}$ and $B=\\var{A2}$ then:

\n

$$
\\begin{aligned}
\\var{k1} A+B &= \\var{k1} \\var{A}+\\var{A2} \\\\
&=\\var{ad1}+\\var{A2} \\\\
&= \\var{ad1a}
\\end{aligned}
$$

\n

b)

\n

If $C=\\var{B}$ and $D=\\var{B2}$ then:

\n

$$
\\begin{aligned}
C - \\var{k2} D &= \\var{B} - \\var{k2} \\var{B2} \\\\
&=\\var{B}-\\var{ad2} \\\\
&= \\var{ad2a}
\\end{aligned}
$$

\n

c)

\n

If $E=\\var{C}$ and $F=\\var{C2}$ then:

\n

$$
\\begin{aligned}
\\var{k3} E - \\var{k1}F &= \\var{k3} \\var{C} - \\var{k1} \\var{C2} \\\\
&=\\var{ad3} - \\var{ad3a} \\\\
&= \\var{ad3b}
\\end{aligned}
$$

\n

d)

\n

If $G=\\var{D}$ and $H=\\var{D2}$ then:

\n

$$
\\begin{aligned}
\\var{k4} G- H &= \\var{k4} \\var{D}+\\var{D2} \\\\
&=\\var{ad4}+\\var{D2} \\\\
&= \\var{ad4a}
\\end{aligned}
$$

\n

e)

\n

If $I=\\var{EE}$ and $J=\\var{EE2}$ then:

\n

$$
\\begin{aligned}
\\var{k5} I-J &= \\var{k5} \\var{EE}-\\var{EE2} \\\\
&=\\var{ad5}-\\var{EE2} \\\\
&= \\var{ad5a}
\\end{aligned}
$$

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Let:

\n

$$
A=\\var{A},  B=\\var{A2}
$$

\n

Calculate $\\var{k1}A + B$:

\n

$\\var{k1} A+B=$[[0]]

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Let:

\n

$$
C=\\var{B}, D=\\var{B2}
$$

\n

Calculate $C-\\var{k2}D$:

\n

$C-\\var{k2}D=$ [[0]]

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Let:

\n

$$
E=\\var{C}, F=\\var{C2}
$$

\n

Calculate $\\var{k3}E-\\var{k1}F$:

\n

$\\var{k3}E-\\var{k1}F=$ [[0]]

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Let:

\n

$$
G=\\var{D}, H=\\var{D2}
$$

\n

Calculate $\\var{k4}G+H$:

\n

$\\var{k4}G+H=$ [[0]]

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Let:

\n

$$
I=\\var{EE}, J=\\var{EE2}
$$

\n

Calculate $\\var{k5}I-J$

\n

$\\var{k5}I-J=$ [[0]]

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