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Aqsa Amir

Grokking Modern System Design Interview for Engineers & Managers

Ace your System Design Interview and take your career to the next level. Learn to handle the design of applications like Netflix, Quora, Facebook, Uber, and many more in a 45-min interview. Learn the RESHADED framework for architecting web-scale applications by determining requirements, constraints, and assumptions before diving into a step-by-step design process.

Breaking the original matrix, **LU factorization**. The product,

The equation can be represented in the matrices as follows:

The following example depicts the result of LU factorization.

Let's find the LU factorization of the following example:

Where

Find the upper triangular matrix

Note:In the Gaussian elimination method,1. All rows containing zeros must be at the bottom of the matrix.

2. The first non-zero entry of every row should be on the right side of the first non-zero entry of the previous row.

Perform the following operation:

The resulting matrix obtained is as follows:

Now, perform another operation to achieve the upper triangular matrix,

The resulting matrix obtained is as follows:

Calculate the lower triangular matrix using the equation

Otherwise,

We will use the second method to clarify it further.

$O_{1}$ made$l_{31}$ zero, hence, we will replace it with the multiplier element 2.$O_{2}$ made$l_{32}$ zero, hence, we will replace it with the multiplier element -4.$l_{21}$ was already zero so it will remain the same.

The resulting matrix is as follows:

- Given that
$AX = C$ , we will replace$A$ with$LU$ since$A=LU$ . - Now in the equation
$LUX=C$ , replace$UX$ with$Y$ . - Solve the system of linear equations using the two equations
$LY = C$ and$UX=Y$ .

Solving the first equation,

The solution matrix,

Solving the second equation,

The solution matrix,

Q

Solve the following systems of equations:

6$x_{1}$ + 18$x_{2}$ + 3$x_{3}$ = 3

2$x_{1}$ + 12$x_{2}$ + $x_{3}$ = 19

4$x_{1}$ + 15$x_{2}$ + 3$x_{3}$ = 0

{3, -9, 12}

{1, -3, 7}

{-3, 3, -11}

{-2, -5, 13}

LU factorization is used in numerous applications such as:

- Finding the inverse of a matrix
- Finding the determinant of a matrix
- Finding current in a circuit
- Solving discrete dynamical system problems

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Grokking Modern System Design Interview for Engineers & Managers

Ace your System Design Interview and take your career to the next level. Learn to handle the design of applications like Netflix, Quora, Facebook, Uber, and many more in a 45-min interview. Learn the RESHADED framework for architecting web-scale applications by determining requirements, constraints, and assumptions before diving into a step-by-step design process.

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