Linear Algebra for Machine Learning and Data Science (4)

Systems of linear equations

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Q1.

Translate the following description into a system of linear equations.

Bob has a mixture of apples and oranges of the same size, which he split into two bags. The first bag contains 4 apples and 3 oranges and weighs 700g. The second bag contains 2 apples and 6 oranges and weighs 1000g.

4a + 3b = 700

2a + 6b = 1000

Q2.

Translate the following system of linear equations into the matrix form:

$$ \begin{cases} 5x + 3y + 5z = 6 \\ 7x + 2y + 8z = 17 \\ 4x + 3z = 8 \end{cases} $$

It's a 3x3 matrix with the coefficients:

5	3	5
7	2	8
4	0	3

Q3.

Compute the determinant of the following matrix:

$$ \left[\begin{array}{ccc}1 & 2 & 1 \\0 & 3 & 5 \\4 & 3 & 8\end{array}\right] $$

37

 

$ = (1 * 3 * 8 + 2 * 5 * 4 + 1 * 0 * 3) - (1 * 3 * 4 + 2 * 0 * 8 + 1 * 5 * 3) $

 

$ = (24 + 40 + 0) - (12 + 0 + 15) $

 

$ = 64 - 27 = 37 $

Q4.

Does the matrix above have any linear dependent row?

No

Q5.

Is the matrix singular or nonsingular?

Non-singular because the determinant is not 0

All the information here is based on the Linear Algebra for Machine Learning and Data Science | Coursera from DeepLearning.AI