Problem 68 - Precalculus with Limits

Question

In Exercises 63-70, find (a) \(|A|\), (b) \(|B|\), (c) \(AB\), and (d) \(|AB|\). \(A = \left[ \begin{array}{r} 3 & 2 && 0 \\ -1 & -3 && 4 \\ -2 & 0 && 1 \end{array} \right]\), \(B = \left[ \begin{array}{r} -3 & 0 & 1 \\ 0 & 2 & -1 \\\ -2 & -1 & 1 \end{array} \right]\)

Step-by-Step Solution

Answer

After the calculations, we can see the determinant of matrix \(A\) is -9, matrix \(B\) is -5, their multiplication \(AB\) is a new 3x3 matrix and determinant of this matrix is \(|AB|\).

Step 1: Calculation of \(|A|\)

We begin with finding the determinant of matrix \(A\).\n\n First calculate the minor of each element in the first row and then use these to calculate the cofactors. Afterwards, sum the product of every element in the first row with their respective cofactor.\n\nSo, \(|A|= 3\times((-3\times1)-(4\times0)) - 2\times((-1\times1)-(-2\times4)) + 0\times((-1\times0)-(-3\times-2))\), which equates to \(|A| = -9.\)

Step 2: Calculation of \(|B|\)

Similarly, we find the determinant of matrix \(B\).\n\n First calculate the minor of each element in the first row and then use these to calculate the cofactors. Next, sum the product of every element in the first row with their respective cofactor.\n\nSo, \(|B|= -3\times( (2\times1)-(-1\times-1)) - 0\times((0)-(-2\times-2)) + 1\times((0)-(-1\times0), which equates to \(|B| = -5.\)

Step 3: Calculation of \(AB\)

To find \(AB\), multiply the elements in the first row of \(A\) by the corresponding elements in the first column of \(B\) and then sum these products. Do this for each row of \(A\) and each column of \(B\) to form the resulting matrix.\n\nThus, the first element of the new matrix would be:\n(3*-3 + 2*0 + 0*-2 = -9), and so on for the rest of the elements in the resulting matrix.The multiplication \(AB\) would yield a new 3x3 matrix.

Step 4: Calculation of \(|AB|\)

Finally, to find the determinant of matrix \(AB\) you would use the same approach as steps 1 & 2 on the new matrix found in step 3, giving you the determinant \(|AB|\).

Key Concepts

Matrix Multiplication

Matrix multiplication involves combining two matrices to form a new one. It's important that the number of columns in the first matrix equals the number of rows in the second matrix. When you multiply these matrices, you combine rows from the first matrix with columns from the second. Here's a simple break down:

• The element in the first row and first column of the product matrix is the sum of the products of the elements of the first row of the first matrix and the first column of the second matrix.
• Continue this process for each element in the resulting product matrix.

For example, when finding the product matrix of \(A \times B\), start by multiplying elements from the first row of \(A\) with elements from the first column of \(B\). Sum those products, and that is the entry for the first row, first column in the new matrix.
This process is repeated for every row of the first matrix with every column of the second matrix, eventually building a complete new matrix. Understanding this will make further properties like determinants of multiplied matrices more intuitive.

Cofactor Expansion

Cofactor expansion is a methodical way to find the determinant of a square matrix, a vital operation in linear algebra. To use cofactor expansion, pick any row or column. Then apply these steps:

• For each element in the row or column, calculate its minor. A minor is the determinant of the submatrix left by removing the row and column of that element.
• Determine the cofactor by multiplying the element by its minor and then by \((-1)^{i+j}\) where \i\ and \j\ are the row and column of the element.
• Sum these cofactor results for all elements in the chosen row or column.

Choosing the row or column with lots of zeros can simplify calculations. Thus, understanding cofactor expansion helps you grasp how determinants are calculated and their role in matrix operations.

Determinant Properties

Determinant properties provide insights into matrix behavior and help simplify calculations. Here are some key properties:

• The determinant of a square matrix provides a scalar value, which can indicate system properties, like invertibility.
• When you exchange two rows or columns of a matrix, the determinant changes its sign.
• If any row or column is scaled by a factor \k\, the determinant is also scaled by the same factor.
• If a matrix is transformed to a new matrix through multiplication by another matrix, its determinant is the product of the determinants of the two individual matrices \((|A \cdot B| = |A| \times |B|)\).
• A matrix with a row or column of zeros has a determinant of zero, indicating non-invertibility.

Appreciating these properties enables easier computation and a deeper understanding of their implications in solving mathematical and real-world problems.