Question

Question: CRITICAL THINKING

Make a conjecture about how you could find the determinant of a $4\times 4$ matrix using the expansion by minors method. Use a diagram in your explanation.

Short answer

A four order determinant is$\left|\begin{array}{cccc}a& b& c& d\\ e& f& g& h\\ i& j& k& l\\ m& n& o& p\end{array}\right|$

For a determinant of order 4, minors are determinant of order 3. On expanding four order determinant by first row minors are

$\left|\begin{array}{ccc}f& g& h\\ j& k& l\\ n& o& p\end{array}\right|,\left|\begin{array}{ccc}e& g& h\\ i& k& l\\ m& o& p\end{array}\right|,\left|\begin{array}{ccc}e& f& h\\ i& j& l\\ m& n& p\end{array}\right|,\text{and}\left|\begin{array}{ccc}e& f& g\\ i& j& k\\ m& n& o\end{array}\right|$

To expand the determinant, multiply the minors with positive or negative sign as shown in the following image

$\left[\begin{array}{cccc}+& -& +& -\\ +& -& +& -\\ +& -& +& -\\ +& -& +& -\end{array}\right]$

In the above image position signs alternate between positive and negative, beginning with a positive sign in first row, first column.

Determinant of order 4 is multiply a row number with its minor and position sign, then add the terms with positive position sign and then subtract negative position sign terms from it as shown below

$\left|\begin{array}{cccc}a& b& c& d\\ e& f& g& h\\ i& j& k& l\\ m& n& o& p\end{array}\right|=a\left|\begin{array}{ccc}f& g& h\\ j& k& l\\ n& o& p\end{array}\right|-b\left|\begin{array}{ccc}e& g& h\\ i& k& l\\ m& o& p\end{array}\right|+c\left|\begin{array}{ccc}e& f& h\\ i& j& l\\ m& n& p\end{array}\right|-d\left|\begin{array}{ccc}e& f& g\\ i& j& k\\ m& n& o\end{array}\right|$

Step-by-step solution

- Write a four order determinant

A four order determinant is determinant of a square matrix of order 4.

It can be written as

$\left|\begin{array}{cccc}a& b& c& d\\ e& f& g& h\\ i& j& k& l\\ m& n& o& p\end{array}\right|$

- Write minors for 4 order determinant

For a determinant of order 4, minors are determinant of order 3. On expanding four order determinant by first row minors are

$\left|\begin{array}{ccc}f& g& h\\ j& k& l\\ n& o& p\end{array}\right|,\left|\begin{array}{ccc}e& g& h\\ i& k& l\\ m& o& p\end{array}\right|,\left|\begin{array}{ccc}e& f& h\\ i& j& l\\ m& n& p\end{array}\right|,\text{and}\left|\begin{array}{ccc}e& f& g\\ i& j& k\\ m& n& o\end{array}\right|$

- Image of position sign 

To expand the determinant, multiply the minors with positive or negative sign as shown in the following image

$\left[\begin{array}{cccc}+& -& +& -\\ +& -& +& -\\ +& -& +& -\\ +& -& +& -\end{array}\right]$

In the above image position signs alternate between positive and negative, beginning with a positive sign in first row, first column.

- Find determinant

Determinant of order 4 is multiply a row number with its minor and position sign, then add the terms with positive position sign and then subtract negative position sign terms from it as shown below

$\left|\begin{array}{cccc}a& b& c& d\\ e& f& g& h\\ i& j& k& l\\ m& n& o& p\end{array}\right|=a\left|\begin{array}{ccc}f& g& h\\ j& k& l\\ n& o& p\end{array}\right|-b\left|\begin{array}{ccc}e& g& h\\ i& k& l\\ m& o& p\end{array}\right|+c\left|\begin{array}{ccc}e& f& h\\ i& j& l\\ m& n& p\end{array}\right|-d\left|\begin{array}{ccc}e& f& g\\ i& j& k\\ m& n& o\end{array}\right|$