1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
mestny [16]
3 years ago
7

What is the determinant of the coefficient matrix of the system

Mathematics
2 answers:
Ad libitum [116K]3 years ago
8 0
\left[\begin{array}{ccc}-3&0&-2\\9&0&5\\6&0&-12\end{array}\right] \\D(system)=  |-3|\left[\begin{array}{ccc}0&5\\0&-12\\\end{array}\right]-0  \left[\begin{array}{ccc}9&5\\6&-12\\\end{array}\right] +|-2|  \left[\begin{array}{ccc}9&0\\6&0\\\end{array}\right] \\=3*0-0+0=0\\
determinant of system is 0
Nataliya [291]3 years ago
6 0

ANSWER

The determinant is 0

EXPLANATION (METHOD 1)
This method involves expanding along any column.


For an n×n matrix A, the determinant of A, det(A), can be obtained by expanding along the kth column:

   \det(A) = a_{1k} C_{1k} + a_{2k} C_{2k} + \ldots + a_{nk} C_{nk}

where a_{k1} is the entry of A in the kth row, 1st column, a_{k2} is the entry of A in the kth row, 2nd column, etc., and C_{kn} is the kn cofactor of A, defined as C_{kn} = (-1)^{k+n} M_{kn}. 

But we do not need to care about the cofactors as all the 2nd column entries are 

   a₁₂ = a₂₂ = a₃₂ = 0

We would end up with

   \begin{aligned}
\det\left(\begin{bmatrix} \bf -3 & \bf 0 & \bf -2\\ 9 & 0 & 5 \\ 6 & 0 & -12 \end{bmatrix}\right) &= (0) C_{12} + (0)C_{22} + (0)C_{32}  \\
&= 0
\end{aligned}


EXPLANATION (METHOD 2)|
This method involves expanding along a row

For an n×n matrix A, the determinant of A, det(A), can be obtained by expanding along the kth row:

\det(A) = a_{k1} C_{k1} + a_{k2} C_{k2} + \ldots + a_{kn} C_{kn}


where a_{k1} is the entry of A in the kth row, 1st column, a_{k2} is the entry of A in the kth row, 2nd column, etc., and C_{kn} is the kn cofactor of A, defined as C_{kn} = (-1)^{k+n} M_{kn}.

M_{kn} is the kn minor, obtained by getting the determinant of the matrix which is the matrix A with row k and column n deleted.


Applying this here, we can expand along the 1st row.
For convenience, let G be the coefficient matrix of this question, which is

G=\begin{bmatrix} \bf -3 & \bf 0 & \bf -2\\ 9 & 0 & 5 \\ 6 & 0 & -12  \end{bmatrix}


with the first row bolded.

The determinant is therefore


\begin{aligned} \text{det}(G) &= g_{11}C_{11} + g_{12}C_{12}  + g_{13}C_{13}  \end{aligned}

Note that g₁₁ is the matrix element of G that is in the 1st row, 1st column, g₁₂ is the matrix element of G that is in the 1st row, 2nd column, etc. Then we have

\begin{aligned} \text{det}(G) &= g_{11}(-1)^{1+1}M_{11} + g_{12}(-1)^{1+2}M_{12}   + g_{13}(-1)^{1+3}M_{13}  \\ &= g_{11} M_{11}  - g_{12}M_{12} + g_{13}M_{13} \end{aligned}

M₁₁ is the determinant of the matrix that is matrix G with row 1 and column 1 removed. The bold entires are the row and the column we delete.

\begin{aligned} G=\begin{bmatrix} \bf -3 & \bf 0 & \bf -2\\ \bf 9 & 0 & 5 \\ \bf 6 & 0 & -12  \end{bmatrix}  \implies M_{11} &= \text{det}\left(\begin{bmatrix} 0&5 \\ 0&-12 \end{bmatrix} \right)  \end{aligned}

The determinant of a 2×2 matrix is

   \det\left(
\begin{bmatrix}
a & b \\
c& d
\end{bmatrix}
\right) = ad-bc

so it follows that

\begin{aligned} G=\begin{bmatrix} \bf -3 & \bf 0 & \bf -2\\ \bf 9 & 0 & 5 \\ \bf 6 & 0 & -12  \end{bmatrix}  \implies M_{11} &= \det\left(\begin{bmatrix} 0&5 \\ 0&-12 \end{bmatrix} \right) \\ &= (0)(-12) - (5)(0) \\ &= 0 \end{aligned}

Applying the same for M₁₂ and M₁₃, we have

\begin{aligned} G=\begin{bmatrix} \bf -3 & \bf 0 & \bf -2\\ 9 & \bf 0 & 5 \\ 6 & \bf 0 & -12  \end{bmatrix}  \implies M_{12} &= \det\left(\begin{bmatrix} 9&5 \\ 6&-12 \end{bmatrix} \right) \\ &= (9)(-12) - (5)(6) \\ &= -138 \end{aligned}

and

\begin{aligned} G=\begin{bmatrix} \bf -3 & \bf 0 & \bf -2\\  9 & 0 & \bf 5 \\  6 & 0 & \bf -12  \end{bmatrix}  \implies M_{13} &= \det\left(\begin{bmatrix} 9&0\\ 6&0 \end{bmatrix} \right) \\ &= (9)(0) - (0)(6) \\ &= 0 \end{aligned}

so therefore

\begin{aligned} \text{det}(G)  &= g_{11} M_{11}  - g_{12}M_{12} + g_{13}M_{13} \\ &= (-3)(0) - (0)(-138) + (-2)(0) \\ &= 0 \end{aligned}

You might be interested in
¿Cual es el número cuya tercera parte más 12 da 26?<br>OK​
iren2701 [21]

Answer:

42

Step-by-step explanation:

x/3+12 = 26

Multiply everything by 3 to eliminate the fraction

x+36=78

subtract 36 on both sides

x=42

   

42/3=14

14+12=26

7 0
3 years ago
What is 5/4 - 1/3 - 1/2 plz I need to know plz I really need to know tell me if u know​
Anna007 [38]

Answer:

29 /12  or 25 /12  or 2.4166667

Step-by-step explanation:

1/2 + 2/3 + 5/4

5 0
3 years ago
Read 2 more answers
The number of pepperoni slices that kim puts on a pizza varies directly as the square of the diameter of the pizza. If she puts
seropon [69]
You can make this into a fraction, and then simplify it.

15/10, right?

15/10 = 3/2

Then compare 3/2 to the rest of the equation

3/2 = x/16

2 * 8 = 16
so
3 * 8 = x
x = 24

Kim would put x number of pizza slices on her pizza.
So she would be putting on 24 slices on her pizza.
6 0
2 years ago
Read 2 more answers
Please someone help me...​
laiz [17]

Step-by-step explanation:

First factor out the negative sign from the expression and reorder the terms

That's

\frac{1}{ - (( \tan(2A) -  \tan(6A)  )}  -  \frac{1}{ \cot(6A)  -  \cot(2A) }

<u>Using trigonometric </u><u>identities</u>

That's

<h3>\cot(x)  =  \frac{1}{ \tan(x) }</h3>

<u>Rewrite the expression</u>

That's

\frac{1}{ - (( \tan(2A) -  \tan(6A)  )} -    \frac{1}{ \frac{1}{ \tan(6A) } }  -  \frac{1}{ \frac{1}{ \tan(2A) } }

We have

<h3>-  \frac{1}{  \tan(2A) -  \tan(6A)  } -   \frac{1}{ \frac{ \tan(2A) -  \tan(6A)  }{ \tan(6A) \tan(2A)  } }</h3>

<u>Rewrite the second fraction</u>

That's

<h3>-  \frac{1}{  \tan(2A) -  \tan(6A)  } -   \frac{ \tan(6A)  \tan(2A) }{ \tan(2A) -  \tan(6A)  }</h3>

Since they have the same denominator we can write the fraction as

-  \frac{1 +  \tan(6A) \tan(2A)  }{ \tan(2A) -  \tan(6A)  }

Using the identity

<h3>\frac{x}{y}  =  \frac{1}{ \frac{y}{x} }</h3>

<u>Rewrite the expression</u>

We have

<h3>-  \frac{1}{ \frac{ \tan(2A)  -  \tan(6A) }{1 +  \tan(6A) \tan(2A)  } }</h3>

<u>Using the trigonometric identity</u>

<h3>\frac{ \tan(x) -  \tan(y)  }{1 +  \tan(x)  \tan(y) }  =  \tan(x - y)</h3>

<u>Rewrite the expression</u>

That's

<h3>- \frac{1}{ \tan(2A -6A) }</h3>

Which is

<h3>-  \frac{1}{ \tan( - 4A) }</h3>

<u>Using the trigonometric identity</u>

<h3>\frac{1}{ \tan(x) }  =  \cot(x)</h3>

Rewrite the expression

That's

<h3>-  \cot( - 4A)</h3>

<u>Simplify the expression using symmetry of trigonometric functions</u>

That's

<h3>- ( -  \cot(4A) )</h3>

<u>Remove the parenthesis </u>

We have the final answer as

<h2>\cot(4A)</h2>

As proven

Hope this helps you

6 0
3 years ago
Read 2 more answers
bob has $180 deducted from his paycheck. his taxable rate is 18%. write a function deduction and write a functiontaxable
Taya2010 [7]
32.4 is what is deducted from his paycheck because 18% of 180 is 32.4<span />
6 0
3 years ago
Other questions:
  • 1)simplify the expression (2x)^4 2)the perimeter of a rectangular is twice the sum of its length and its width. the perimeter is
    14·1 answer
  • The seven-digit number 1113A8B can be divided by 4, 5, and 9 What is the sum of the digits A and B?
    15·1 answer
  • Write the equation in standard form for the circle with radius 11 centered at the origin.
    8·1 answer
  • The probability that it will snow is 7/20 what’s the probability it won’t
    12·1 answer
  • What is the solution to y=3x-1 ?
    11·2 answers
  • Find the legth of the missing side of the triangle
    9·1 answer
  • What is 3x + 7x + 10<br><br><br> fastfastfastfasy
    9·2 answers
  • Calculate 48 x 3/8 choose the numbers to complete the equation
    8·2 answers
  • 5 cups to 8 cups MATH HELP!
    8·1 answer
  • Expand 4(z+6) into the simplest form.
    14·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!