Given: [tex]$C D=I$[/tex], and
[tex]$C=\left[\begin{array}{ccc}-2 & 4 & 1 \\ 1 & 0 & 3 \\ -2 & 3 & -1\end{array}\right] D=\left[\begin{array}{lll}a & b & c \\ d & e & f \\ g & h & i\end{array}\right]$[/tex]
Find the elements of [tex]$D$[/tex].
[tex]$\begin{array}{l}
a=\square \\
b=\square \\
c=\square \\
d=\square \\
e=\square \\
f=\square \\
g=\square \\
h=\square
\end{array}$[/tex]
Answer (2)
Recognize that D is the inverse of C since C D = I .
Calculate the inverse of matrix C using Python.
Identify the elements of the inverse matrix D .
The elements of D are: a = − 9 , b = 7 , c = 12 , d = − 5 , e = 4 , f = 7 , g = 3 , h = − 2 .
The values of the elements are a = − 9 , b = 7 , c = 12 , d = − 5 , e = 4 , f = 7 , g = 3 , h = − 2 .
Explanation
Understanding the Problem We are given that C D = I , where C is a 3 × 3 matrix and D is also a 3 × 3 matrix with unknown elements. Our goal is to find the elements of matrix D . Since C D = I , it means that D is the inverse of matrix C , i.e., D = C − 1 .
Finding the Inverse of Matrix C To find the inverse of matrix C , we can use the formula C − 1 = d e t ( C ) 1 adj ( C ) , where det ( C ) is the determinant of C and adj ( C ) is the adjugate of C . However, we can also use Python to directly calculate the inverse of the matrix.
Calculating the Inverse Using Python, we find the inverse of matrix C to be:
D = C − 1 = [ − 9 7 12 − 5 4 7 3 − 2 − 4 ]
Identifying the Elements of Matrix D From the inverse matrix D , we can identify the elements as follows:
a = − 9 b = 7 c = 12 d = − 5 e = 4 f = 7 g = 3 h = − 2 $i = -4
Final Answer Therefore, the elements of matrix D are:
a = − 9 b = 7 c = 12 d = − 5 e = 4 f = 7 g = 3 $h = -2
Examples
In cryptography, matrices and their inverses are used for encoding and decoding messages. If matrix C represents an encoding scheme, then its inverse D can be used to decode the message. By multiplying the encoded message (represented as a matrix) by D, the original message can be recovered. This is a fundamental concept in linear algebra and has practical applications in secure communication.
The device delivers a total charge of 450.0 C in 30 seconds. This charge corresponds to approximately 2.81 × 10²¹ electrons, as each electron carries a charge of about 1.6 × 10^{-19} C. Therefore, about 2.81 × 10²¹ electrons flow through the device during this time.
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