Determinants and inverses of 2 x 2 matrices

Welcome to Determinants and Inverses!

In your journey through the Vectors, matrices and transformations section, you have already learned how to add and multiply matrices. Now, we are going to look at two powerful tools that help us "unlock" matrices: the Determinant and the Inverse.

Think of the Inverse as the "undo" button for matrix multiplication. Just like the opposite of multiplying by \(5\) is multiplying by \(\frac{1}{5}\), matrices have their own way of reversing an operation. To find that "undo" button, we first need to calculate a special number called the Determinant.

Note: For this course, you only need to know how to do this for \(2 \times 2\) matrices (matrices with 2 rows and 2 columns).

1. The Determinant

The determinant is a single number calculated from a square matrix. It tells us a lot about the matrix's properties, especially how it transforms shapes.

The Formula

If we have a matrix \(M = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\), the determinant is calculated as:

\(\text{Determinant of } M = ad - bc\)

Step-by-Step Guide:
1. Multiply the top-left number \(a\) by the bottom-right number \(d\).
2. Multiply the top-right number \(b\) by the bottom-left number \(c\).
3. Subtract the second result from the first.

Example:

Find the determinant of \(A = \begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix}\).

\(a = 3, b = 2, c = 1, d = 4\)

\(\text{Determinant} = (3 \times 4) - (2 \times 1)\)
\(\text{Determinant} = 12 - 2 = 10\)

Quick Tip: Think of it as a "Criss-Cross" subtraction. Multiply the main diagonal, then subtract the product of the other diagonal.

2. The Inverse of a Matrix

The inverse of a matrix \(M\) is written as \(M^{-1}\). When you multiply a matrix by its inverse, you get the Identity Matrix \(I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\).

In simple terms: \(M \times M^{-1} = I\)

The Formula (On your Formula Sheet!)

Don't worry about memorizing this perfectly, as it is provided in your exam! For a matrix \(M = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\):

\(M^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}\)

This looks like a lot, so let's break it down into simple movements.

How to construct the Inverse:

Step 1: Calculate the determinant (\(ad - bc\)).
Step 2: Swap the positions of \(a\) and \(d\) (the main diagonal).
Step 3: Change the signs of \(b\) and \(c\) (make them negative if they were positive, or positive if they were negative).
Step 4: Multiply the whole thing by \(1 \div \text{determinant}\).

Did you know? If the determinant is \(0\), you can't divide by it, which means the matrix doesn't have an inverse! However, for Specification B, you will focus on non-singular matrices (matrices that do have an inverse).

3. Putting it all together: A Worked Example

Find the inverse of \(B = \begin{pmatrix} 4 & 5 \\ 2 & 3 \end{pmatrix}\).

Step 1: Find the determinant
\((4 \times 3) - (5 \times 2) = 12 - 10 = 2\)

Step 2: Swap \(a\) and \(d\)
The \(4\) and \(3\) trade places: \(\begin{pmatrix} 3 & \dots \\ \dots & 4 \end{pmatrix}\)

Step 3: Change signs of \(b\) and \(c\)
The \(5\) becomes \(-5\) and the \(2\) becomes \(-2\): \(\begin{pmatrix} 3 & -5 \\ -2 & 4 \end{pmatrix}\)

Step 4: Put it together
\(B^{-1} = \frac{1}{2} \begin{pmatrix} 3 & -5 \\ -2 & 4 \end{pmatrix}\)

You can leave your answer like this, or multiply the \(\frac{1}{2}\) into the matrix:
\(B^{-1} = \begin{pmatrix} 1.5 & -2.5 \\ -1 & 2 \end{pmatrix}\)

Common Pitfalls to Avoid

• Mixing up the swap: Remember, you only swap the numbers on the "downward" diagonal (\(a\) and \(d\)). The other two (\(b\) and \(c\)) stay where they are but change signs.
• Negative Number Errors: Be very careful when calculating the determinant if \(b\) or \(c\) are already negative.
  Example: If \(bc = -6\), then \(ad - bc\) becomes \(ad - (-6)\), which is \(ad + 6\).
• Forgetting the \(1/\text{det}\) part: Finding the new matrix is only half the job. You must multiply by the reciprocal of the determinant!

Key Takeaways for Exam Success

Quick Review:

• Determinant: \(ad - bc\)
• Inverse Process: Swap \(a\) and \(d\), negate \(b\) and \(c\), multiply by \(1/\text{det}\).
• Check your work: If you have time, multiply your original matrix by your inverse. If you get \(\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\), you are 100% correct!
• Formula Sheet: Always double-check the formula sheet during the exam to ensure you haven't swapped the wrong letters.

Prerequisite Check: If you are struggling with the multiplication part, review the "Matrix arithmetic" chapter to see how scalars (the number outside the matrix) interact with the numbers inside.