Problem 39 - Algebra and Trigonometry Real Mathematics, Real People

Question

Finding the Product of Two Matrices Find \(A B,\) if possible. $$A=\left[\begin{array}{r} 5 \\ -3 \\ 4 \end{array}\right], \quad B=\left[\begin{array}{lll} 2 & -8 & 4 \end{array}\right]$$

Step-by-Step Solution

Answer

The product of matrix A and matrix B is the matrix \(C = \[ \[10, -40, 20\], \[-6, 24, -12\], \[8, -32, 16\] \]\).

Step 1: Check if matrix multiplication is possible

Check that the number of columns in the first matrix, matrix A, is equal to the number of rows in the second matrix, matrix B. In this case, matrix A has 1 column and matrix B has 1 row, so their multiplication is possible.

Step 2: Set up the resultant matrix

Since matrix A is a 3x1 matrix and matrix B is a 1x3 matrix, their product will be a 3x3 matrix. Let's denote the entries of the resultant matrix as \(C_{ij}\), where \(i\) represents the row number and \(j\) represents the column number.

Step 3: Compute the entries of the resultant matrix

Calculate each entry, \(C_{ij}\), in the resultant matrix by multiplying the corresponding entries in the row of matrix A and column of matrix B. This gives us: \(C_{11} = (5)(2) = 10\), \(C_{12} = (5)(-8) = -40\), \(C_{13} = (5)(4) = 20\), \(C_{21} = (-3)(2) = -6\), \(C_{22} = (-3)(-8) = 24\), \(C_{23} = (-3)(4) = -12\), \(C_{31} = (4)(2) = 8\), \(C_{32} = (4)(-8) = -32\), \(C_{33} = (4)(4) = 16\).

Step 4: Form the resultant matrix

After computing the entries of the resultant matrix, place them into their respective positions to form the resultant matrix: \(C = \[ \[10, -40, 20\], \[-6, 24, -12\], \[8, -32, 16\] \]\).

Key Concepts

Resultant Matrix

Matrix multiplication is a fascinating operation that results in a new matrix, known as the resultant matrix.

This new matrix summarizes all the unique interactions between rows of the first and columns of the second matrix.

In our exercise, by multiplying matrices A and B, we obtained a 3x3 resultant matrix, which tells us how every element of matrix A interacts with matrix B. This is represented in the grid-like structure where each entry or element inside denotes a specific calculation of these interactions.

Each entry in this 3x3 grid is calculated meticulously to portray the comprehensive information distilled from both matrices being multiplied.

Matrix Dimensions

In matrix multiplication, understanding the dimensions is crucial because they determine if the multiplication is possible.

Dimensions are expressed as 'rows' x 'columns'. For instance, Matrix A in our problem is a 3x1 matrix, meaning it has 3 rows and 1 column. On the contrary, Matrix B is a 1x3 matrix, indicating it has 1 row and 3 columns.

For multiplication to occur, the number of columns in the first matrix must be equal to the number of rows in the second. Here, both satisfy the condition, allowing multiplication. This compatibility enables us to compute the 3x3 resultant matrix, determined by the external dimensions of the matrices (3 from matrix A and 3 from matrix B).

Matrix Entries

Each entry within a matrix holds a vital piece of the puzzle, representing specific components of matrix calculations.

When two matrices are multiplied, each entry in the resultant matrix is derived from the sum of products of corresponding elements in the involved row and column.

• For example, the entry in the first row first column of the resultant matrix, denoted as \(C_{11}\), is calculated by multiplying the elements in the first row of matrix A with the elements in the first column of matrix B: \((5)(2) = 10\).
• Similarly, each subsequent entry \(C_{ij}\) requires analogous calculations; for instance, \(C_{21}\) or the entry in the second row first column is \((-3)(2) = -6\), and so forth.

This systematic approach continues until all entries of the resultant matrix are calculated, painting the complete picture of matrix multiplication.