Problem 40 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find \((a)|A|,(b)|B|,(c) A B,\) and \((d)|A B| .\) What do you notice about \(|A B| ?\) $$A=\left[\begin{array}{rr}4 & 0 \\\3 & -2\end{array}\right], \quad B=\left[\begin{array}{ll}-1 & 1 \\\\-2 & 2\end{array}\right]$$

Step-by-Step Solution

Answer

The determinants of matrices A and B are -8 and -2 respectively. The product of A and B is a new 2x2 matrix and its determinant is 24. It can be noticed that \(|AB|\) is not equal to \(|A| * |B|\), which is -16, demonstrating that the determinant of the product of two matrices is not necessarily the product of their determinants.

Step 1: Calculation of |A|

To calculate the determinant (|A|) of matrix A, subtract the product of the elements of the second diagonal from the product of the elements of the main diagonal. \(|A| = (4*-2) - (0*3) = -8.\)

Step 2: Calculation of |B|

Apply same procedure for matrix B. \(|B| = (-1*2) - (1*-2)= -4+2 = -2.\)

Step 3: Calculation of AB

Matrix multiplication is done element by element in a cross manner. The first element in the first row of the resulting matrix is found by multiplying the first element in the first row of the first matrix by the first element in the first column of the second matrix and then adding the multiplication of the second element in the first row of the first matrix by the second element in the first column of the second matrix. These steps are repeated for all the rows and columns of the matrices A and B. Then, \(AB = \left[\begin{array}{rr}(-4) & 4 \\(-7) & 1\end{array}\right].\)

Step 4: Calculation of |AB|

To find determinant of the resultant matrix AB, same calculation as in step 1 and 2 is carried out. \ \(|AB| = (-4*1) - (4*-7)= -4 + 28 = 24.\)

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra, where we combine two matrices to produce a third matrix. This operation requires the number of columns in the first matrix to be equal to the number of rows in the second matrix. Here's how you multiply matrices:

• Take each row element of the first matrix and multiply it by the corresponding column element of the second matrix.
• Sum these products to get the element of the new matrix.

In our example above, we calculate the product of two matrices, \(A\) and \(B\) to get \(AB\). The elements are calculated using a cross manner, meaning each element of the resulting matrix comes from combining the row of the first matrix with the column of the second matrix. For example, the first element is computed by multiplying and adding the elements as explained in step 3.
This results in a new matrix: \[AB = \begin{bmatrix} -4 & 4 \-7 & 1 \end{bmatrix}\]

Determinant Calculation

Determinant calculation is a key concept in understanding matrices and their properties. The determinant of a matrix is a scalar value that provides important insights into the matrix characteristics.For a 2x2 matrix, the determinant is calculated with the formula:\[|A| = ad - bc\]Here, \(a, b, c,\) and \(d\) represent the elements of the matrix:\[A = \begin{bmatrix} a & b \c & d \end{bmatrix}\]This means you multiply the elements of the main diagonal and subtract the product of the elements of the other diagonal. In our example matrix \(A\), the determinant \(|A| = (4 \times (-2)) - (0 \times 3) = -8\)
This method is also applied to calculate \(|B|\). Remember, determinant calculation also plays a crucial role in identifying invertibility of a matrix, which is determined by whether the determinant is non-zero.

Square Matrices

Square matrices are matrices with an equal number of rows and columns, such as 2x2, 3x3, etc. They are central in matrix operations because many mathematical concepts, like determinants and eigenvalues, are only defined for square matrices.A noteworthy property of square matrices is that they have a unique determinant, a number that can reveal certain characteristics about the matrix. For instance, if a square matrix has a determinant of zero, it indicates the matrix is singular, meaning it doesn't have an inverse.
The matrices \(A\) and \(B\) in the exercise are both 2x2 square matrices. Thus, we can calculate their determinants and multiply them by each other. Matrix multiplication is simpler within square matrices, following standard rules of combining rows and columns.

Properties of Determinants

Determinants have several important properties that are valuable in matrix analysis. Understanding these helps in simplifying complex linear algebra problems.

• Multiplicative Property: The determinant of a product of two matrices is equal to the product of their determinants. For matrices \(A\) and \(B\), \(|AB| = |A| \times |B|\).
• Zero and Non-zero Determinant: A zero determinant indicates a singular matrix, which has no inverse, and a system of linear equations represented by the matrix has either no solution or infinitely many solutions.
• Swap Rows or Columns: Swapping any two rows or columns of a matrix multiplies the determinant by \(-1\).

In our exercise, even though both \(|A|\) and \(|B|\) are negative, multiplying them still yields \((|A| \times |B| = 16)\), which interestingly differs from \(|AB| = 24\), showing how cross-products and sign changes impact the result.