Problem 1 - Algebra 2

Question

Show that the matrices are multiplicative inverses. $$ \left[\begin{array}{rr}{3} & {2} \\ {4} & {3}\end{array}\right],\left[\begin{array}{rr}{3} & {-2} \\ {-4} & {3}\end{array}\right] $$

Step-by-Step Solution

Answer

After matrix multiplication, the product of the given matrices is the identity matrix, proving that they are multiplicative inverses.

Step 1: Write down the given matrices

Identify the two given matrices. We will call them matrix A and matrix B.Matrix A: \[\begin{pmatrix}3 & 2 \4 & 3\end{pmatrix}\]Matrix B:\[\begin{pmatrix}3 & -2 \-4 & 3\end{pmatrix}\]

Step 2: Write the formula for matrix multiplication

For two 2x2 matrices A and B, the multiplication is done as follows:\[AB = \begin{pmatrix}a_{11} & a_{12} \a_{21} & a_{22}\end{pmatrix} \begin{pmatrix}b_{11} & b_{12} \b_{21} & b_{22}\end{pmatrix} = \begin{pmatrix}a_{11}b_{11} + a_{12}b_{21} & a_{11}b_{12} + a_{12}b_{22} \a_{21}b_{11} + a_{22}b_{21} & a_{21}b_{12} + a_{22}b_{22}\end{pmatrix}\]

Step 3: Multiply matrix A by matrix B

Perform the multiplication using the elements of matrices A and B.\[AB = \begin{pmatrix}3 & 2 \4 & 3\end{pmatrix} \begin{pmatrix}3 & -2 \-4 & 3\end{pmatrix} = \begin{pmatrix}3(3) + 2(-4) & 3(-2) + 2(3) \4(3) + 3(-4) & 4(-2) + 3(3)\end{pmatrix} \]\[= \begin{pmatrix}9 - 8 & -6 + 6 \12 - 12 & -8 + 9\end{pmatrix} \]\[= \begin{pmatrix}1 & 0 \0 & 1\end{pmatrix} \]

Step 4: Verify the result is the identity matrix

Check if the result of the matrix multiplication AB is the 2x2 identity matrix.\[AB = \begin{pmatrix}1 & 0 \0 & 1\end{pmatrix}\]The result is indeed the 2x2 identity matrix, which confirms that the given matrices A and B are multiplicative inverses of each other.

Key Concepts

Matrix Multiplication

When you multiply two matrices, it's quite different from multiplying plain numbers. Imagine it as a systematic dance where each element of the rows of the first matrix interacts with elements of the columns of the second matrix to produce a new set of numbers.

In technical terms, when you have two 2x2 matrices - let's call them Matrix A and Matrix B - you find their product by taking the sum of the products of the elements in the rows of Matrix A with the corresponding elements in the columns of Matrix B. This might sound confusing, but once you apply the formula for each element of the resulting matrix, it's like following a recipe. You'll always end up with a new 2x2 matrix that's a blend of both the matrices you started with.

For instance, if A consists of elements, say \(a_{11}\), \(a_{12}\), \(a_{21}\), and \(a_{22}\), and B consists of elements \(b_{11}\), \(b_{12}\), \(b_{21}\), and \(b_{22}\), the product \(AB\) will have each element calculated as follows: the top-left element is \(a_{11}b_{11} + a_{12}b_{21}\), and so on. Remember, though, matrix multiplication is not commutative, meaning \(AB\) does not always equal \(BA\).

Identity Matrix

The identity matrix is like the number 1 in the land of matrices. Just like multiplying any number by 1 leaves the number unchanged, when any matrix - let's call it Matrix X - is multiplied by an identity matrix, it stays the same, it's the unchanged, unshaken Matrix X.

In terms of its layout, the identity matrix is a special square matrix that has ones \(1\) on the diagonal (the top-left to the bottom-right corner) and zeros everywhere else, which makes it look a bit like a model runway with a straight path of ones. For the 2x2 case, it's simply a matrix with \(1\) and \(0\) in a neat layout: \[\begin{pmatrix}1 & 0 \: & :\0 & 1\end{pmatrix}\]. This simplicity hides a super power - it's the key that confirms whether two matrices are multiplicative inverses of each other. If you multiply two matrices together and end up with this simple, yet powerful identity matrix, you've hit the jackpot - you've found multiplicative inverses!

Algebraic Operations on Matrices

Algebra is not just about x's and y's; with matrices, it's a whole new game with its own rules. One of these rules is that you can do addition, subtraction, and yes, multiplication. However, you can't just toss the numbers around as you might with algebraic terms.

In matrix land, addition and subtraction are fairly straightforward – they're like a polite conversation where corresponding elements from each matrix meet and either combine forces by adding up or go their separate ways by subtracting. But multiplication? That's the intricate tango dance we talked about in the earlier section.

Oh! And division? In the matrix world, division is not allowed. Instead, we work with something called the inverse which, if you multiply by the original matrix, gives you the matrix equivalent of 1, the identity matrix. But remember, not all matrices can bust these moves; for instance, they all have to be the same size to add or subtract them, and to multiply, the number of columns in the first one has to match the number of rows in the second. Play by these rules, and you'll be the algebraic maestro of the matrix symphony!

2x2 Matrices

2x2 matrices are the cool kids on the basic block of matrix algebra. Picture a small square with four numbers – that's your classic 2x2 matrix. The interesting thing about these matrices is that they're quite easy to work with when it comes to finding inverses compared to their larger companions.

While larger matrices require sophisticated tactics to find their inverses, the 2x2 matrices are more accessible and the process to find their multiplicative inverses is more direct. Just remember a key formula involving determinants and adjugates, and you've got your inverse recipe.

Speaking of being direct, operations with 2x2 matrices provide the most immediate and clear examples of how matrix arithmetic works, such as the multiplication process we've used to show that two matrices are inverses of each other. So starting with 2x2 matrices is a wonderful way to ease into the universe of matrix algebra before diving into the deep end with bigger and more complex matrix structures.