Nativity Girl’s Catholic School Mathematics test 2 for Grade 11C&D Students from
Instruction one: Choose the correct answer from the given alternatives.
21. If and , then which one of the following is the value of such that ?
A.
B.
C.
D.
2. If is singular matrix then the value of k is
A. 2
B.
C. 3
D
3. Let and . If and then what is the solution of the system ?
A.
B.
C.
D.
4. What is the determinant of ?
A. 20
B. 0
C. 16
D. 24
5. If A is a matrix with , then what is the ?
A.
B. 12
C. 36
D. 48
Instruction Two: Write the correct answer in the blank space provided below.
6. If , then and .
7. If is skew symmetric matrix then the value of
Instruction Three: workout (show necessary steps).
8. What should be the values of a and so that the system solution.
9. Find the inverse of (if it exists) using adjoint method?
10. Suppose and be matrices such that and , then find the
None of the options match this result, so let’s check the options again. The closest option is: B. .
Problem 2
To find such that is singular, we need to set the determinant of to zero.
Calculate the determinant of :
Set the determinant to zero:
The answer is: D. .
Problem 3
We need to solve the system where .
Use Cramer’s Rule:
Calculate :
Since , we can find using the formula:
Find the adjugate of :
The adjugate can be calculated, but we need to find directly.
Substituting values:
We can substitute values from the options into the equation to find the correct solution.
After testing the options, we find: A. is the correct solution.
Problem 4
Calculate the determinant of :
Using the determinant formula for a matrix:
Calculating:
The answer is: B. 0.
Problem 5
Given , the determinant remains the same:
The answer is: B. 12.
Problem 6
We have the matrix equation:
Setting up the equations:
From :
Substituting into :
Substituting into :
The answers are:
( x = 1, y = 2,
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The Deep Dive
If you think matrices are just collections of numbers, think again! The historical journey of matrices began back in ancient China, where they were first used for solving systems of linear equations, long before we had the term “matrix.” The groundbreaking work of mathematicians like Arthur Cayley and Hermann Grassmann in the 19th century transformed matrices into the powerful tools they are today, opening doors to advancements in engineering, computer science, and physics!
Now, let’s talk about how we actually use these mathematical constructs in our everyday lives! From algorithms that power social media feeds to video game graphics and financial modeling, matrices are everywhere! They help in image processing, representing data in machine learning, and even in cryptography to secure your online transactions. So, the next time you solve a math problem involving matrices, remember you’re tapping into something that’s crucial in our tech-driven world!