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Any idea how to compute the determinant of $4 \times 4$ matrix $A$ when

\begin{equation} A = \begin{bmatrix} 1 & 4 & 8 & 1\\ 0 & 30 & 1 & 0 \\ 0 & 2 & 0 & 0 \\ 1 & 2 & 9 & a \end{bmatrix}. \end{equation}

The $a$ in the $A_{44}$ location is really confusing me!

littleO
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Phil
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  • I've computed the value of the determinant using Matlab and it came out to 2-2*a. I don't quite understand what you mean when you ask how to compute it. Certainly computing determinants of 3x3 matrices by hand is very tedious. I would imagine for 4x4 matrices to be doubly more so! Can't you use computer software to find the determinant? – Malcolm Nov 09 '15 at 14:31
  • @Malcolm Note that the determinant is much easier to compute when we expand down the fourth column. – littleO Nov 09 '15 at 14:52
  • I understand your problem concerning the symbol $a$. Usually, we use numbers in calculations. $a$ is just standing in for a number. Imagine $a=1234.56$ and you had to rewrite it every time you're writing the determinant of $A$. I don't know about you, but I'd rather write $a$ each time. At the end, if you wanted a numerical answer, you could replace $a$ with its value. The only thing you have to worry about $a$ is that it could be equal to 0. So don't divide by $a$ in your calculations. – MasB Nov 09 '15 at 15:19

2 Answers2

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Just expand across the third row, as @Surb suggested: \begin{equation} |A| = -2 \left| \begin{pmatrix} 1 & 8 & 1\\ 0 & 1 & 0 \\ 1 & 9 & a \end{pmatrix} \right| =-2 \left| \begin{pmatrix} 1 & 1 \\ 1 & a \end{pmatrix} \right| = -2(a - 1) = 2 - 2a. \end{equation}

Another option is to expand down the fourth column:

\begin{align} | A| &= -1 \left|\begin{pmatrix} 0 & 30 & 1 \\ 0 & 2 & 0 \\ 1 & 2 & 9 \end{pmatrix} \right| + a \left | \begin{pmatrix} 1 & 4 & 8 \\ 0 & 30 & 1 \\ 0 & 2 & 0 \end{pmatrix} \right |. \end{align}

The two determinants on the right simplify greatly by expanding down the first column: \begin{equation} \left|\begin{pmatrix} 0 & 30 & 1 \\ 0 & 2 & 0 \\ 1 & 2 & 9 \end{pmatrix} \right| = 1\left| \begin{pmatrix} 30 & 1 \\ 2 & 0 \end{pmatrix} \right| = -2 \end{equation} and \begin{equation} \left | \begin{pmatrix} 1 & 4 & 8 \\ 0 & 30 & 1 \\ 0 & 2 & 0 \end{pmatrix} \right | = 1 \left| \begin{pmatrix} 30 & 1 \\ 2 & 0 \end{pmatrix} \right| = -2. \end{equation} So we see that $|A| = 2 - 2a$.

littleO
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In your case, the determinant of A is a function of a.

Here is how to solve 4x4 matrix https://www.sophia.org/tutorials/finding-the-determinant-of-a-4x4-matrix--5

but when you encounter a, you should keep it untouched. You can still use basic algebra to keep equations simple. For example, (3*a + 4) + (4*a + 5) = 7a+9 etc.

Here is Maple doing the calculation for you:

with(LinearAlgebra): M:=Matrix(4,[[1,4,8,1],[0,30,1,0],[0,2,0,0],[1,2,9,a]]); Determinant(M);
-2 a + 2

What you actually need to calculate in general case of 4x4 matrices is:

M:=Matrix(4,[[a11,a12,a13,a14],[a21,a22,a23,a24],[a31,a32,a33,a34],[a41,a42,a43,a44]]);
Determinant(M);

which yields: 

a11 a22 a33 a44 - a11 a22 a34 a43 - a11 a23 a32 a44 + a11 a23 a34 a42 + a11 a24 a32 a43 - a11 a24 a33 a42 - a12 a21 a33 a44 + a12 a21 a34 a43 + a12 a23 a31 a44 - a12 a23 a34 a41 - a12 a24 a31 a43 + a12 a24 a33 a41 + a13 a21 a32 a44 - a13 a21 a34 a42 - a13 a22 a31 a44 + a13 a22 a34 a41 + a13 a24 a31 a42 - a13 a24 a32 a41 - a14 a21 a32 a43 + a14 a21 a33 a42 + a14 a22 a31 a43 - a14 a22 a33 a41 - a14 a23 a31 a42 + a14 a23 a32 a41

here you can see this rule https://en.wikipedia.org/wiki/Determinant#n.C2.A0.C3.97.C2.A0n_matrices used.

Mikael Kuisma
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