Table of Contents

System of Linear Equations

This example shows you how to solve a system of linear equations in Excel. For example we have the following system of linear equations:

5x + 1y + 8z = 46
4x – 2y = 12
6x + 7y + 4z = 50

In matrix notation this can be written as AX = B

<!– A
<td width=”69″>

<!– X
<td width=”15″>

<!– B
<td width=”15″>

<!– A
<td align=”left”>with A =

<!– X
<td>

<!– B
<td>

<!– A
<td>

<!– X
<td>

<!– B
<td>

5 1 8 x 46
4 -2 0 X = y B = 12
6 7 4 z 50

If A-1 (the inverse of A) exists we can multiply both sides by A-1 to obtain X = A-1B. To solve this system of linear equations in Excel execute the following steps.

1. Use the MINVERSE function to return the inverse matrix of A. First select the range B6:D8. Next insert the MINVERSE function shown below. Finish by pressing CTRL + SHIFT + ENTER.

MINVERSE function

Note: the formula bar indicates that the cells contain an array formula. Therefore you cannot delete a single result. To delete the results select the range B6:D8 and press Delete.

2. Use the MMULT function to return the product of matrix A-1 and B. First select the range G6:G8. Next insert the MMULT function shown below. Finish by pressing CTRL + SHIFT + ENTER.

MMULT function

3. Put it all together. First select the range G6:G8. Next insert the formula shown below. Finish by pressing CTRL + SHIFT + ENTER.

Solution

4. If you have Excel 365 or Excel 2021 simply select cell G6 enter the same formula as above and press Enter. Bye bye curly braces.

Dynamic Array Formula

Note: this dynamic array formula entered into cell G6 fills multiple cells. Wow! This behavior in Excel 365/2021 is called spilling.

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