A 4 by 4 determinant of a 4 by 4 matrix can be solved in a similar way to solving 3 by 3 matrix. Consider the following 4 by 4 matrix.

2a + b + c + 3d = 4 ---->(1)

4a - 2b + d = 0 --->(2)

-a + b + 2d = 20 -----(3)

3b + c - d = 13 ---->(4)

where a, b, c and d are variables.

Now the coefficients of the equations can be written in the determinant form D as follows.

Now we can calculate the determinant using either the row as the starting basis,

or use a column as the basis,

Let say we use the first row. Then expand the determinant D as follows,

To understand how this is written see the figure below,

And the sign are alternating.

Then again you can utilize this expansion trick to expand the 3 by 3 matrix as shown below,

Now you can evaluate this equation easily. The answer to this equation is below.

That is the determinant is -16.

2a + b + c + 3d = 4 ---->(1)

4a - 2b + d = 0 --->(2)

-a + b + 2d = 20 -----(3)

3b + c - d = 13 ---->(4)

where a, b, c and d are variables.

Now the coefficients of the equations can be written in the determinant form D as follows.

Now we can calculate the determinant using either the row as the starting basis,

or use a column as the basis,

Let say we use the first row. Then expand the determinant D as follows,

To understand how this is written see the figure below,

And the sign are alternating.

Then again you can utilize this expansion trick to expand the 3 by 3 matrix as shown below,

Now you can evaluate this equation easily. The answer to this equation is below.

That is the determinant is -16.

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