Solved on Jan 25, 2024

Find the determinant of the matrix product of $A=\left[\begin{array}{ll}4 & 2\end{array}\right]$ and $B=\left[\begin{array}{c}-1 \\ 3\end{array}\right]$ .

STEP 1

Assumptions
1. Matrix $A$ is given by $A=\left[\begin{array}{ll}4 & 2\end{array}\right]$
2. Matrix $B$ is given by $B=\left[\begin{array}{c}-1 \\ 3\end{array}\right]$
3. We need to find the determinant of the product $BA$ , denoted as $|BA|$

STEP 2

First, we need to multiply matrix $B$ by matrix $A$ . The product of a $2 \times 1$ matrix and a $1 \times 2$ matrix will result in a $2 \times 2$ matrix.
$BA = B \times A$

STEP 3

Perform the matrix multiplication.
$BA = \left[\begin{array}{c}-1 \\ 3\end{array}\right] \left[\begin{array}{ll}4 & 2\end{array}\right]$

STEP 4

Multiply each element of matrix $B$ by each element of matrix $A$ and sum the products to find the elements of the resulting matrix.
$BA = \left[\begin{array}{cc} -1 \times 4 & -1 \times 2 \\ 3 \times 4 & 3 \times 2 \end{array}\right]$

STEP 5

Calculate the elements of the resulting matrix.
$BA = \left[\begin{array}{cc} -4 & -2 \\ 12 & 6 \end{array}\right]$

STEP 6

Now we need to find the determinant of the resulting $2 \times 2$ matrix $BA$ .
$|BA| = \left|\begin{array}{cc} -4 & -2 \\ 12 & 6 \end{array}\right|$

STEP 7

The determinant of a $2 \times 2$ matrix is calculated by subtracting the product of the elements on the secondary diagonal from the product of the elements on the main diagonal.
$|BA| = (-4)(6) - (-2)(12)$

STEP 8

Calculate the determinant.
$|BA| = -24 - (-24)$

STEP 9

Simplify the expression.
$|BA| = -24 + 24$

STEP 10

Calculate the final value of the determinant.
$|BA| = 0$
The determinant of the product $BA$ is 0, which corresponds to option d.

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Find the determinant of the matrix product of $A=\left[\begin{array}{ll}4 & 2\end{array}\right]$ and $B=\left[\begin{array}{c}-1 \\ 3\end{array}\right]$ .

STEP 1

Assumptions
1. Matrix $A$ is given by $A=\left[\begin{array}{ll}4 & 2\end{array}\right]$
2. Matrix $B$ is given by $B=\left[\begin{array}{c}-1 \\ 3\end{array}\right]$
3. We need to find the determinant of the product $BA$ , denoted as $|BA|$

STEP 2

First, we need to multiply matrix $B$ by matrix $A$ . The product of a $2 \times 1$ matrix and a $1 \times 2$ matrix will result in a $2 \times 2$ matrix.
$BA = B \times A$

STEP 3

Perform the matrix multiplication.
$BA = \left[\begin{array}{c}-1 \\ 3\end{array}\right] \left[\begin{array}{ll}4 & 2\end{array}\right]$

STEP 4

Multiply each element of matrix $B$ by each element of matrix $A$ and sum the products to find the elements of the resulting matrix.
$BA = \left[\begin{array}{cc} -1 \times 4 & -1 \times 2 \\ 3 \times 4 & 3 \times 2 \end{array}\right]$

STEP 5

Calculate the elements of the resulting matrix.
$BA = \left[\begin{array}{cc} -4 & -2 \\ 12 & 6 \end{array}\right]$

STEP 6

Now we need to find the determinant of the resulting $2 \times 2$ matrix $BA$ .
$|BA| = \left|\begin{array}{cc} -4 & -2 \\ 12 & 6 \end{array}\right|$

STEP 7

The determinant of a $2 \times 2$ matrix is calculated by subtracting the product of the elements on the secondary diagonal from the product of the elements on the main diagonal.
$|BA| = (-4)(6) - (-2)(12)$

STEP 8

Calculate the determinant.
$|BA| = -24 - (-24)$

STEP 9

Simplify the expression.
$|BA| = -24 + 24$

STEP 10

Calculate the final value of the determinant.
$|BA| = 0$
The determinant of the product $BA$ is 0, which corresponds to option d.

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Find the determinant of the matrix product of A=[42]A=\left[\begin{array}{ll}4 & 2\end{array}\right]A=[4​2​] and B=[−13]B=\left[\begin{array}{c}-1 \\ 3\end{array}\right]B=[−13​].

STEP 1

Assumptions1. Matrix AAA is given by A=[42]A=\left[\begin{array}{ll}4 & 2\end{array}\right]A=[4​2​]2. Matrix BBB is given by B=[−13]B=\left[\begin{array}{c}-1 \\ 3\end{array}\right]B=[−13​]3. We need to find the determinant of the product BABABA, denoted as ∣BA∣|BA|∣BA∣

STEP 2

First, we need to multiply matrix BBB by matrix AAA. The product of a 2×12 \times 12×1 matrix and a 1×21 \times 21×2 matrix will result in a 2×22 \times 22×2 matrix.BA=B×ABA = B \times ABA=B×A

STEP 3

Perform the matrix multiplication.BA=[−13][42]BA = \left[\begin{array}{c}-1 \\ 3\end{array}\right] \left[\begin{array}{ll}4 & 2\end{array}\right]BA=[−13​][4​2​]

STEP 4

Multiply each element of matrix BBB by each element of matrix AAA and sum the products to find the elements of the resulting matrix.BA=[−1×4−1×23×43×2]BA = \left[\begin{array}{cc} -1 \times 4 & -1 \times 2 \\ 3 \times 4 & 3 \times 2 \end{array}\right]BA=[−1×43×4​−1×23×2​]

STEP 5

Calculate the elements of the resulting matrix.BA=[−4−2126]BA = \left[\begin{array}{cc} -4 & -2 \\ 12 & 6 \end{array}\right]BA=[−412​−26​]

STEP 6

Now we need to find the determinant of the resulting 2×22 \times 22×2 matrix BABABA.∣BA∣=∣−4−2126∣|BA| = \left|\begin{array}{cc} -4 & -2 \\ 12 & 6 \end{array}\right|∣BA∣=∣∣​−412​−26​∣∣​

STEP 7

The determinant of a 2×22 \times 22×2 matrix is calculated by subtracting the product of the elements on the secondary diagonal from the product of the elements on the main diagonal.∣BA∣=(−4)(6)−(−2)(12)|BA| = (-4)(6) - (-2)(12)∣BA∣=(−4)(6)−(−2)(12)

STEP 8

Calculate the determinant.∣BA∣=−24−(−24)|BA| = -24 - (-24)∣BA∣=−24−(−24)

STEP 9

Simplify the expression.∣BA∣=−24+24|BA| = -24 + 24∣BA∣=−24+24

STEP 10

Calculate the final value of the determinant.∣BA∣=0|BA| = 0∣BA∣=0The determinant of the product BABABA is 0, which corresponds to option d.

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Find the determinant of the matrix product of $A=\left[\begin{array}{ll}4 & 2\end{array}\right]$ and $B=\left[\begin{array}{c}-1 \\ 3\end{array}\right]$ .

Assumptions
1. Matrix $A$ is given by $A=\left[\begin{array}{ll}4 & 2\end{array}\right]$
2. Matrix $B$ is given by $B=\left[\begin{array}{c}-1 \\ 3\end{array}\right]$
3. We need to find the determinant of the product $BA$ , denoted as $|BA|$

First, we need to multiply matrix $B$ by matrix $A$ . The product of a $2 \times 1$ matrix and a $1 \times 2$ matrix will result in a $2 \times 2$ matrix.
$BA = B \times A$

Perform the matrix multiplication.
$BA = \left[\begin{array}{c}-1 \\ 3\end{array}\right] \left[\begin{array}{ll}4 & 2\end{array}\right]$

Multiply each element of matrix $B$ by each element of matrix $A$ and sum the products to find the elements of the resulting matrix.
$BA = \left[\begin{array}{cc} -1 \times 4 & -1 \times 2 \\ 3 \times 4 & 3 \times 2 \end{array}\right]$

Calculate the elements of the resulting matrix.
$BA = \left[\begin{array}{cc} -4 & -2 \\ 12 & 6 \end{array}\right]$

Now we need to find the determinant of the resulting $2 \times 2$ matrix $BA$ .
$|BA| = \left|\begin{array}{cc} -4 & -2 \\ 12 & 6 \end{array}\right|$

The determinant of a $2 \times 2$ matrix is calculated by subtracting the product of the elements on the secondary diagonal from the product of the elements on the main diagonal.
$|BA| = (-4)(6) - (-2)(12)$

Calculate the determinant.
$|BA| = -24 - (-24)$

Simplify the expression.
$|BA| = -24 + 24$

Calculate the final value of the determinant.
$|BA| = 0$
The determinant of the product $BA$ is 0, which corresponds to option d.