Q46.

Question

Question: CRITICAL THINKING

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

Step-by-Step Solution

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Answer

A four order determinant is abcdefghijklmnop

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

 fghjklnop,eghiklmop,efhijlmnp, and efgijkmno

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

 ++++++++

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

abcdefghijklmnop=afghjklnopbeghiklmop+cefhijlmnpdefgijkmno

1Step 1 - Write a four order determinant

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

It can be written as

 abcdefghijklmnop

2Step 2 - 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

 fghjklnop,eghiklmop,efhijlmnp, and efgijkmno

3Step 3 - Image of position sign

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

 ++++++++

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

4Step 4 - 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

 abcdefghijklmnop=afghjklnopbeghiklmop+cefhijlmnpdefgijkmno