Let A=[12−110311−1] i) Find the determinant of A ii)Verify that the matrix [−34146410−11414−12] is the inverse matrix of A

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Reggie

2021-02-14

Let

A = [\begin{matrix} 1 & 2 & - 1 \\ 1 & 0 & 3 \\ 1 & 1 & - 1 \end{matrix}]

i) Find the determinant of A
ii)Verify that the matrix

[\begin{matrix} - \frac{3}{4} & \frac{1}{4} & \frac{6}{4} \\ 1 & 0 & - 1 \\ \frac{1}{4} & \frac{1}{4} & - \frac{1}{2} \end{matrix}]

is the inverse matrix of A

jlo2niT

Skilled2021-02-15Added 96 answers

Step 1
For part (i) of a:
According to the given information, it is required to calculate the determinant of matrix

A = [\begin{matrix} 1 & 2 & - 1 \\ 1 & 0 & 3 \\ 1 & 1 & - 1 \end{matrix}]

For this use the formula

| \begin{matrix} x_{1} & y_{1} & z_{1} \\ x_{2} & y_{2} & z_{2} \\ x_{3} & y_{3} & z_{3} \end{matrix} | = x_{1} | \begin{matrix} y_{2} & z_{2} \\ y_{3} & z_{3} \end{matrix} | - y_{1} | \begin{matrix} x_{2} & z_{2} \\ x_{3} & z_{3} \end{matrix} | + z_{1} | \begin{matrix} x_{2} & y_{2} \\ x_{3} & y_{3} \end{matrix} |

Step 2
Using above formula determinant of A can be calculated as

| A | = | \begin{matrix} 1 & 2 & - 1 \\ 1 & 0 & 3 \\ 1 & 1 & - 1 \end{matrix} |

= 1 | \begin{matrix} 0 & 3 \\ 1 & - 1 \end{matrix} | - 2 | \begin{matrix} 1 & 3 \\ 1 & - 1 \end{matrix} | - 1 | \begin{matrix} 1 & 0 \\ 1 & 1 \end{matrix} |

= 1 (0 \times - 1 - 3 \times 1) - 2 (1 \times - 1 - 3 \times 1) - 1 (1 \times 1 - 0 \times 1)

=1(-3)-2(-1-3)-1
=-3+8-1
=4
Thus, determinant of matrix A is 4.
Step 3
For part (ii) of a:
To verify that the matrix

[\begin{matrix} - \frac{3}{4} & \frac{1}{4} & \frac{6}{4} \\ 1 & 0 & - 1 \\ \frac{1}{4} & \frac{1}{4} & - \frac{1}{2} \end{matrix}]

is inverse of given matrix A use the definition of identity matrix that states:

A A^{- 1} = I

Step 4
Now calculate the multiplication of these two matrices, as

[\begin{matrix} 1 & 2 & - 1 \\ 1 & 0 & 3 \\ 1 & 1 & - 1 \end{matrix}] [\begin{matrix} - \frac{3}{4} & \frac{1}{4} & \frac{6}{4} \\ 1 & 0 & - 1 \\ \frac{1}{4} & \frac{1}{4} & - \frac{1}{2} \end{matrix}] = [\begin{matrix} 1 \times - \frac{3}{4} + 2 \times 1 - 1 \times \frac{1}{4} & 1 \times \frac{1}{4} + 2 \times 0 - 1 \times \frac{1}{4} & 1 \times \frac{6}{4} + 2 \times - 1 - 1 \times - \frac{1}{2} \\ 1 \times - \frac{3}{4} + 0 \times 1 + 3 \times \frac{1}{4} & 1 \times \frac{1}{4} + 0 \times 0 + 3 \times \frac{1}{4} & 1 \times \frac{6}{4} + 0 \times - 1 + 3 \times - \frac{1}{2} \\ 1 \times - \frac{3}{4} + 1 \times 1 - 1 \times \frac{1}{4} & 1 \times \frac{1}{4} + 1 \times 0 - 1 \times \frac{1}{4} & 1 \times \frac{6}{4} + 1 \times - 1 - 1 \times - \frac{1}{2} \end{matrix}]

This is helpful 19 Flag

Jeffrey Jordon

Expert2022-01-29Added 2605 answers

Answer is given below (on video)

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