Math Problem Statement

Consider the matrices $\mathbf{A}, \mathbf{B}, \mathbf{C}$ and $\mathbf{D}$ indicated by the output grids below, as well as the shaded green areas in each grid: [asy] size(500); //Draws grid lines with the given parameters: void gridLines(int xmin, int xmax, int ymin, int ymax) { for(int i = ymin+1; i< ymax; ++i) { draw((xmin,i)--(xmax,i), mediumgray); } for(int i = xmin+1; i< xmax; ++i) { draw((i,ymax)--(i,ymin), mediumgray); } } //Gives the maximum line that fits in the box. path maxLine(pair A, pair B, real xmin, real xmax, real ymin, real ymax) { real s = abs(xmax) + abs(xmin) + abs(ymax) + abs(ymin) + abs(A.x - xmin) + abs(A.x-xmax) + abs(A.y - ymin) + abs(A.y - ymax); pair[] endpoints = intersectionpoints(A+2sunit(B-A) -- A-2sunit(B-A), (xmin, ymin)--(xmin, ymax)--(xmax, ymax)--(xmax, ymin)--cycle); if (endpoints.length >= 2) return endpoints[0]--endpoints[1]; else return nullpath; } pair O, O2, O3, O4, AI, AJ, BI, BJ, CI, CJ, DI, DJ; pair tailAI, tailAJ, tailBI, tailBJ, tailCI, tailCJ, tailDI, tailDJ; path shapeA, shapeB, shapeC, shapeD; AI = dir(50); AJ = 2dir(200); BI = 2dir(20); BJ =dir(-70); CI = sqrt(3)dir(20); CJ =sqrt(6)dir(65); DI = 2dir(-30); DJ = 4dir(-60); //Change this to scale the grid: int big = 5; O = (0,0); O2 = (2big + 2, 0); O3 = (4big + 4, 0); O4 = (6big + 6, 0); //Matrix A output grid: shapeA = -4AI - 3AJ -- -4AI + AJ -- -AI+AJ -- -AI - 3AJ --cycle; fill(shapeA, green); for (int i = -15; i < 15; ++i) { draw(maxLine(iAI, iAI + AJ, -big, big, -big,big), linewidth(0.4)); draw(maxLine(iAJ, iAJ + AI, -big, big, -big, big), linewidth(0.4)); } draw(tailAI --tailAI + AI, red+linewidth(0.8), Arrow(7)); draw(tailAJ-- tailAJ + AJ, blue+linewidth(0.8), Arrow(7)); label("$\mathbf{A} \mathbf{i}$", AI, N, UnFill); label("$\mathbf{A} \mathbf{j}$", AJ, NW, UnFill); //Matrix B output grid: shapeB = -2BI - 3BJ -- -2BI - BJ -- BI- BJ -- BI - 3BJ --cycle; fill(shift(O2)shapeB, green); for (int i = -10; i <10; ++i) { draw(shift(O2)maxLine(iBI, iBI + BJ, -big, big, -big,big), linewidth(0.4)); draw(shift(O2)maxLine(iBJ, iBJ + BI, -big, big, -big, big), linewidth(0.4)); } draw(shift(O2)(tailBI --tailBI + BI), red+linewidth(0.8), Arrow(7)); draw(shift(O2)(tailBJ-- tailBJ + BJ), blue+linewidth(0.8), Arrow(7)); label("$\mathbf{B} \mathbf{i}$", shift(O2)BI, E, UnFill); label("$\mathbf{B} \mathbf{j}$", shift(O2)BJ, S, UnFill); //Matrix C output grid: shapeC = shift(-CI)(-CI -- O-- 2CJ -- 2CJ + CI -- -CJ + CI -- -CJ - CI -- cycle); fill(shift(O3)shapeC, green); for (int i = -10; i <10; ++i) { draw(shift(O3)maxLine(iCI, iCI + CJ, -big, big, -big,big), linewidth(0.4)); draw(shift(O3)maxLine(iCJ, iCJ + CI, -big, big, -big, big), linewidth(0.4)); } draw(shift(O3)(tailCI --tailCI + CI), red+linewidth(0.8), Arrow(7) ); draw(shift(O3)(tailCJ-- tailCJ + CJ), blue+linewidth(0.8), Arrow(7) ); label("$\mathbf{C} \mathbf{i}$", shift(O3)CI, 2E, UnFill); label("$\mathbf{C} \mathbf{j}$", shift(O3)CJ, 2E, UnFill); //Matrix D output grid: shapeD = +DI -- O-- DJ -- DJ - DI -- -DJ - DI -- -DJ + DI -- cycle; fill(shift(O4)shapeD, green); for (int i = -10; i <10; ++i) { draw(shift(O4)maxLine(iDI, iDI + DJ, -big, big, -big,big), linewidth(0.4)); draw(shift(O4)maxLine(iDJ, iDJ + DI, -big, big, -big, big), linewidth(0.4)); } draw(shift(O4)(tailDI --tailDI + DI), red+linewidth(0.8), Arrow(7) ); draw(shift(O4)*(tailDJ-- tailDJ + DJ), blue+linewidth(0.8), Arrow(7) ); label("$\mathbf{D} \mathbf{i}$", shift(O4)*DI, E, UnFill); label("$\mathbf{D} \mathbf{j}$", shift(O4)*DJ, SE, UnFill); [/asy] If each shaded green area is equal to $2$, what are [| \det (\mathbf{A})|, |\det (\mathbf{B})|, |\det (\mathbf{C})|, |\det (\mathbf{D})|]in that order?

Solution

To solve for the determinants of matrices A\mathbf{A}, B\mathbf{B}, C\mathbf{C}, and D\mathbf{D}, we will use the relationship between the area of the parallelogram formed by the vectors corresponding to the matrix and the absolute value of the determinant of the matrix.

Step 1: Understanding the Geometric Interpretation

Each matrix X\mathbf{X} can be represented by its column vectors Xi\mathbf{X} \mathbf{i} and Xj\mathbf{X} \mathbf{j}, where i\mathbf{i} and j\mathbf{j} are the standard basis vectors in 2D space. The parallelogram formed by these vectors has an area equal to ∣det⁡(X)∣|\det(\mathbf{X})|.

Given that each shaded green area (the area of the parallelogram) is equal to 2, we have: ∣det⁡(A)∣=2,∣det⁡(B)∣=2,∣det⁡(C)∣=2,∣det⁡(D)∣=2|\det(\mathbf{A})| = 2, \quad |\det(\mathbf{B})| = 2, \quad |\det(\mathbf{C})| = 2, \quad |\det(\mathbf{D})| = 2

Step 2: Analyze Each Matrix

Matrix A\mathbf{A}

The vectors Ai\mathbf{A} \mathbf{i} and Aj\mathbf{A} \mathbf{j} define a parallelogram, and the area of this parallelogram is given by: ∣det⁡(A)∣=∣Ai×Aj∣|\det(\mathbf{A})| = |\mathbf{A} \mathbf{i} \times \mathbf{A} \mathbf{j}| Given that this area is 2, we have: ∣det⁡(A)∣=2|\det(\mathbf{A})| = 2

Matrix B\mathbf{B}

Similarly, the vectors Bi\mathbf{B} \mathbf{i} and Bj\mathbf{B} \mathbf{j} define a parallelogram with area: ∣det⁡(B)∣=∣Bi×Bj∣|\det(\mathbf{B})| = |\mathbf{B} \mathbf{i} \times \mathbf{B} \mathbf{j}| Given that this area is also 2, we have: ∣det⁡(B)∣=2|\det(\mathbf{B})| = 2

Matrix C\mathbf{C}

The vectors Ci\mathbf{C} \mathbf{i} and Cj\mathbf{C} \mathbf{j} define a parallelogram with area: ∣det⁡(C)∣=∣Ci×Cj∣|\det(\mathbf{C})| = |\mathbf{C} \mathbf{i} \times \mathbf{C} \mathbf{j}| Given that this area is 2, we have: ∣det⁡(C)∣=2|\det(\mathbf{C})| = 2

Matrix D\mathbf{D}

Finally, the vectors Di\mathbf{D} \mathbf{i} and Dj\mathbf{D} \mathbf{j} define a parallelogram with area: ∣det⁡(D)∣=∣Di×Dj∣|\det(\mathbf{D})| = |\mathbf{D} \mathbf{i} \times \mathbf{D} \mathbf{j}| Given that this area is 2, we have: ∣det⁡(D)∣=2|\det(\mathbf{D})| = 2

Final Answer

The absolute values of the determinants of the matrices are: ∣det⁡(A)∣=2,∣det⁡(B)∣=2,∣det⁡(C)∣=2,∣det⁡(D)∣=2|\det(\mathbf{A})| = 2, \quad |\det(\mathbf{B})| = 2, \quad |\det(\mathbf{C})| = 2, \quad |\det(\mathbf{D})| = 2

Would you like more details or have any questions?

Here are 5 related questions:

  1. How do you calculate the determinant of a 2x2 matrix?
  2. Why is the area of the parallelogram formed by two vectors equal to the determinant?
  3. What happens to the determinant if the area of the parallelogram is zero?
  4. How do you interpret the sign of a determinant geometrically?
  5. How does the determinant relate to the volume in 3D for a 3x3 matrix?

Tip: The determinant of a matrix provides both the scaling factor of the area (or volume in higher dimensions) and the orientation of the transformation defined by the matrix.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Linear Algebra
Determinants
Matrix Operations

Formulas

-

Theorems

-

Suitable Grade Level

Advanced High School

Related Recommendation

Determinants of Matrices A, B, C, D, and F, and Inverse Determinant of Matrix D

Find the Area of a Parallelogram Using Determinants: Vectors [3, 2] and [1, 4]

Matrix Operations: Multiplication, Determinants, and Inverses for A and B

Calculate det(C) for Matrix Multiplication of A and B

Area of Parallelogram Generated by Vectors in R^n Using Determinants