3-How to solve for x-y by L-U Decomposition?

Last Updated on September 17, 2025 by Maged kamel

How to solve for x-y by L-U Decomposition?

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The next slide shows the content of post 3- Numeric Linear.

Illustration using the Gauss elimination method for the U matrix formation.

We have a matrix A with two rows and two columns as (2 3 3 4). How to develop the upper matrix U for matrix A?

We use Gauss elimination through elementary row operations to convert the matrix A into an upper matrix and later, use the back substitution to solve for the values of unknowns.

For matrix A 2×2, which is given as (2 3 3 4). To convert this matrix into an upper matrix, we want the diagonal to be non-zero and let  a21 be zero.

Using a11 as a pivot, divide a21/a11 by the minus sign by R1, and add the result to row R2. The second row  R2 will be changed.

We will have a new second row as (0- ½). As we have noticed, there was no operation in the first row of matrix A, which is the same as in the upper matrix. Element U22, second row /2nd column is equal to (a22-(a12/a11).

What about the Lower Matrix? We have by definition L11=L22=1 and L12=0. While for L21 it is equal to (a21/a11), considering that a11 is a pivot. Place the result as a new second row, we can get a zero value for U21, the first element in the second row and first column.

If we want to check whether  L*U  multiplication will lead us to ( 2 3 3 4), we can use row-column multiplication. (1*2+0*0)=2, and (1*3+(0*-1/2)=3. (3/2*2+1*0) =3+0=3 which we will lead us to 3 and (3/2 * 3+1*(-1/2)=4.50-0.50=4. We have a valid L*U  through a valid multiplication. We will use the same matrix A to solve for the values of two unknowns, x and y.

Now, the A matrix as the product of L*U is shown by using the matrix symbols.

Can we use an elementary matrix to get the L and U matrices?

We can get the values of L and U matrices for the given matrix A, but first, we get the U matrix by multiplying matrix A by (-3/2* by the first row) and adding to the second row, and we do the same for the identity matrix. The elementary matrix will be (1 0, -1.5 1). Please refer to the following slide.

We can prove that the elementary matrix is the inverse of the lower matrix L.

We get the inverse of the elementary matrix, the value of L is (1 0, 1.5 1) according to Doolittle’s method.

Write an expression of (L*C)=b as the first step to developing an expression to solve for x-y for two equations-L-U.

Consider the two simultaneous linear equations. The two equations can be expressed by using matrices as A*X=b, where A is a 2×2 matrix for our example, Matrix X is a ( 2×1) matrix that contains (X Y), the unknown variables, while C is a matrix (2×1) of value (13 18).

We use matrix C as an intermediate stage to solve for x-y by L-U Decomposition.

Since we have expressed that matrix A is a product of(L*U), we can further write a new expression as (L*U)*X=b.
The values of L& U and B are known; the only unknown is the column vector matrix X.

A new expression is (U*x)=C, C is a new item that is unknown. Finally, A*x=b can be expressed as(L)*C=b. To solve for a value of C, an inverse matrix of L-1 is to be presented and multiplied by the sides of the given equation. That process will enable us to get a value for matrix C. The modified expression is shown in the next slide image.

Find the inverse of the L matrix as the second step to solve for x-y by L-U decomposition.

In the lower matrix  L, the element is (1  0, +3/2 1), then its inverse of the lower matrix is L-1, we can estimate the determinant value of l, which is =1-0=1.

We divide the adjoint matrix by the determinant. We will swap and make a change of sign of (3/2) to (-3/2), and( 0) will be the same. The final value of L-1 can be expressed as a matrix of 2×2 and is = ( 1 0 -3/2 1).

Get the value of C as the third step to develop an expression to solve for x-y by L-U decomposition.

Get the value of C as the third step to develop an expression to solve for x-y by L-U decomposition. The matrix C is the product of (L-1)*(B).

Since both values are known, we can then get the value of C Multiplying two matrices(2×2) by(2×1) will yield a new matrix with (2×1) with a value of (13 -1.5). The details of the estimation is shown in the next slide image.

Get the value of X as the fourth step to solve for x-y by L-U decomposition.

Get the value of X as the fourth step to solve for x-y f by L-U decomposition. Remember that we have the expression of U*x=C, get an inverse matrix of C, which is a new matrix of 2×2, the X=(U-1)*X, then the final value of X is( 2 3), the matrix (X Y) is a (2×1) matrix. The value of x=2 and y=3.

Substitute the values of x and y to check whether the x,y values are right or wrong.

After checking the two given equations, we find that the estimated values of x and Y satisfy the two equations. We have x = 2 &  y = 3. In the last step, we would like to check if our solution is correct. 2x+3*y=13 and + 3x + 4 y=18.

We introduce x value as=2 and y value as 3, then (2*2+3*3)=13, which will give us the right-hand side as 13.

While 3*2+4 *3=6+12=18, which is already equal to the right-hand side of the second equation.

A quicker way to get X and Y values.

We can express X as equal to the product of the inverse of U by the inverse of L by matrix B. Please refer to the next slide for more details. This method is considered quick if you have the inverse of L and U in hand.

We use the elementary matrix to get the upper matrix U, which equals the product of E1 by matrix A. We can get the inverse matrix of U, as we can see from the next slide image. The elementary matrix can be considered as the L inverse.

The following slide shows the final value of X and Y.

The next post: step-by-step on how to derive the expression for L-U values for-3×3 matrix.

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