Problem 6 - Precalculus: Functions and Graphs

Question

Without expanding, explain why the statement is true. $$\left|\begin{array}{lll} 2 & 1 & 6 \\ 4 & 3 & 3 \\ 2 & 1 & 3 \end{array}\right|=6\left|\begin{array}{lll} 1 & 1 & 2 \\ 2 & 3 & 1 \\ 1 & 1 & 1 \end{array}\right|$$

Step-by-Step Solution

Answer

The left matrix's first row is 2 times the first row of the right matrix.

Step 1: Analyze the Determinant Scalar Multiplication Rule

In determinants, if one row or column of a matrix is multiplied by a scalar, the determinant of the resulting matrix is equal to the original determinant multiplied by that scalar. This is one of the known properties of determinants.

Step 2: Verify the Scalar in the Given Matrices

Looking at the given matrices, compare the first row of the matrix on the left, \([2, 1, 6]\), with the first row of the matrix on the right, \([1, 1, 2]\). Notice that the first row of the left matrix is equivalent to the first row of the right matrix, each element multiplied by 2.

Step 3: Apply the Scalar Multiplication Rule

Since only the first row of the matrix on the left is multiplied by a scalar of 2 when compared with the corresponding first row of the matrix on the right, the entire determinant of the matrix on the left can be expressed as 2 times the determinant of another matrix where the first row is not scaled.

Step 4: Interpret Scaling of the Internal Matrix

After factoring out 2 from the first matrix, compare the resulting internal matrix with the one on the right. Notice that multiplying \(3\) to each element of the adjusted matrix obtained by factoring 2 from the left matrix results in the matrix on the right. Hence, the determinant of the right matrix is indeed present as a part of the determinant of the left matrix, specifically: \(3 \times (2 \times \text{Determinant of the right matrix}) = 6 \times \text{Determinant of the right matrix}.\)

Key Concepts

Scalar Multiplication in Determinants

The concept of scalar multiplication in determinants is a handy tool. When a row or a column of a matrix is multiplied by a scalar value, the determinant of that matrix is multiplied by the same scalar. In simple terms, this means that the determinant scales with the row or column. This rule helps simplify problems and make calculations more manageable without losing any accuracy.
This can be particularly useful when verifying equivalence between two determinants, as in the given exercise. If you multiply only one row, the entire determinant will get scaled by the multiplier too. It's as if the determinant "absorbs" the scalar effect, allowing us to equate matrices, which might initially seem different by their appearance.

Matrix Row Multiplication

Understanding matrix row multiplication is pivotal when working with determinants. When observing matrices, any row can be multiplied by a scalar as an operation. The beauty of this operation lies in how it impacts the determinant: straightforward multiplication of the scalar with the determinant itself.
For instance, consider two matrices where one has a row that's a multiple of another. By identifying this multiplicative relationship, you simplify the original matrix's row to make calculations or comparisons easier. Just like shown in the exercise, one row difference can lead to an understanding of the overall determinant value through multiplication relationships.

• Scalar multiples affect only the row or column multiplied.
• This directly alters the determinant, making understanding this operation essential.
• It's an efficient way to compare matrices and solve determinant-related problems.

Equivalence of Determinants

In matrices, equivalence of determinants refers to the conditions under which two determinants can be considered the same. Often, they are equivalent if you can transform one matrix into another using row or column operations that abide by determinant properties.
The exercise shows us an example of this: by recognizing scalars in the row of a matrix, you can reach equivalence without directly computing these values. By understanding that matrices with different appearances can have equivalent determinants through these transformations, you're equipped to solve determinant equations efficiently.

• Equivalence doesn't require direct calculation; understanding transformation rules suffices.
• Row and column operations facilitate equivalence identification.
• Scalar products can bridge the gap between different matrices.

Mastering this concept facilitates greater efficiency and comprehension in matrix analysis.