Problem 47 - Algebra and Trigonometry

Question

Show that \(B\) is the multiplicative inverse of \(A,\) where $$ A=\left[\begin{array}{ll} {2} & {3} \\ {1} & {2} \end{array}\right] \text { and } B=\left[\begin{array}{rr} {2} & {-3} \\ {-1} & {2} \end{array}\right] $$

Step-by-Step Solution

Answer

B matrix is the multiplicative inverse of matrix A, as the product of A and B, in both orders (AB and BA), results in an Identity matrix.

Step 1: Multiplying A and B

Firstly, multiply two matrices A and B. For instance, proceed with AB. Apply the rules of matrix multiplication where an element in the i-th row and j-th column of the resulting matrix is the sum of the products of elements in the i-th row of the first matrix and j-th column of the second matrix. Here is how to compute AB: AB = \(\left[\begin{array}{ll} {2*2+3*(-1)} & {2*(-3)+3*2} \\ {1*2+2*(-1)} & {1*(-3)+2*2} \end{array}\right]\)

Step 2: Calculating AB

Upon calculating the values in each cell, the resultant matrix AB will be: AB = \(\left[\begin{array}{rr} 4 - 3 & (-6) + 6 \\ 2 - 2 & (-3) + 4 \end{array}\right] = \left[\begin{array}{rr} 1 & 0 \\ 0 & 1 \end{array}\right]\)

Step 3: Multiplying B and A

Next, proceed to compute the multiplication of B and A (BA). Apply the same rules of matrix multiplication as in Step 1: BA = \(\left[\begin{array}{ll} {2*2+(-3)*1} & {2*3+(-3)*2} \\ {(-1)*2+2*1} & {(-1)*3+2*2} \end{array}\right]\)

Step 4: Calculating BA

Compute the values in each cell to get the final BA matrix: BA = \(\left[\begin{array}{rr} 4 - 3 & 6 - 6 \\ (-2) + 2 & (-3) + 4 \end{array}\right] = \left[\begin{array}{rr} 1 & 0 \\ 0 & 1 \end{array}\right]\)

Step 5: Compare AB and BA to Identity matrix

Lastly, compare both products AB and BA with the identity matrix. If both AB and BA are equivalent to the identity matrix, then it proves that B is indeed the multiplicative inverse of A.

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation where the elements of matrices are combined to form a new matrix. Suppose we have two matrices, A and B, and we're calculating their product AB. You must follow specific rules:

• For each element in the product matrix, take the row from matrix A and the column from matrix B.
• Multiply corresponding elements together and sum them up.
• Place that sum at the corresponding position in the result matrix.

To understand this better, consider AB as a product of matrices mentioned in the problem:\[ AB = \left( \begin{array}{cc} 2 & 3 \ 1 & 2 \end{array} \right) \times \left( \begin{array}{cc} 2 & -3 \ -1 & 2 \end{array} \right) \] Compute step-by-step following the rules mentioned, and, as you do so, keep track of how each element interacts with others. This comprehensive process will give clarity in resulting matrices and their transformations during multiplication.

Identity Matrix

The identity matrix is a special type of matrix that acts like the number 1 in matrix multiplication. It means when you multiply any matrix by the identity matrix, it will remain unchanged. The simplest identity matrix is the 2x2 matrix \[ \begin{array}{cc} 1 & 0 \ 0 & 1 \end{array} \] Features of an identity matrix include:

• It is a square matrix (same number of rows and columns).
• The elements along the main diagonal are 1.
• All other elements are 0.

In the problem above, both products AB and BA turned out to match the identity matrix:\[ \left( \begin{array}{cc} 1 & 0 \ 0 & 1 \end{array} \right) \] This property confirmed that B behaves like the multiplicative inverse of A.

Matrix Algebra

Matrix Algebra extends the principles of basic algebra to matrices. This algebra allows us to perform operations such as addition, subtraction, multiplication, and finding inverses with matrices. Key rules and properties include:

• Associative Law: (AB)C = A(BC)
• Distributive Law: A(B+C) = AB + AC
• Commutative Law generally doesn't apply: AB ≠ BA for most matrices

In our exercise, though, we observed both AB and BA results aligning to the identity matrix, which is a unique scenario demonstrating the inverse property specific to matrices. The step-by-step calculations provided clarity on how matrix algebra rules come into play during manipulations.

Invertible Matrix

An invertible matrix is one that has an inverse, meaning there exists another matrix that, when multiplied with it, results in the identity matrix. In mathematical terms, if A is invertible, there exists a matrix B such that: \[ AB = BA = I \] Where I is the identity matrix. Understanding the concept of invertibility is critical because not all matrices have inverses. For example, a matrix is invertible if and only if its determinant is non-zero. In our exercise, matrix A is invertible because multiplying it by B gives the identity matrix, as shown in matrix products both as AB and BA. Always remember:

• Invertibility implies each row is linearly independent from the others.
• It implies there is a unique solution to matrix equations like Ax = b.

These aspects make invertible matrices crucial in solving linear equations and various applications in mathematical computations.