Math Problem Statement

Match the matrix A on the left with the correct expression on the right.

Solution

To match the matrices with the correct expressions, I'll go through the process of finding determinants and inverses, where applicable.

Matrix 1:

A=(315022657)A = \begin{pmatrix} 3 & 1 & 5 \\ 0 & 2 & 2 \\ 6 & 5 & 7 \end{pmatrix} We'll first calculate the determinant:

Det(A)=3225710267+50265\text{Det}(A) = 3 \begin{vmatrix} 2 & 2 \\ 5 & 7 \end{vmatrix} - 1 \begin{vmatrix} 0 & 2 \\ 6 & 7 \end{vmatrix} + 5 \begin{vmatrix} 0 & 2 \\ 6 & 5 \end{vmatrix} Det(A)=3(2725)1(0726)+5(0526)\text{Det}(A) = 3(2 \cdot 7 - 2 \cdot 5) - 1(0 \cdot 7 - 2 \cdot 6) + 5(0 \cdot 5 - 2 \cdot 6) =3(1410)1(12)+5(12)= 3(14 - 10) - 1(-12) + 5(-12) =34+1260=12+1260=36= 3 \cdot 4 + 12 - 60 = 12 + 12 - 60 = -36

So, this matrix corresponds to A. Det A = -36.

Matrix 2:

A=(021321321)A = \begin{pmatrix} 0 & -2 & 1 \\ 3 & -2 & 1 \\ 3 & 2 & -1 \end{pmatrix} The determinant of this matrix is: Det(A)=02121(2)3131+13232\text{Det}(A) = 0 \cdot \begin{vmatrix} -2 & 1 \\ 2 & -1 \end{vmatrix} - (-2) \cdot \begin{vmatrix} 3 & 1 \\ 3 & -1 \end{vmatrix} + 1 \cdot \begin{vmatrix} 3 & -2 \\ 3 & 2 \end{vmatrix} However, after doing the calculations, the determinant turns out to be zero. Therefore, this matrix is singular, and its inverse does not exist. This matrix corresponds to C. The inverse of the matrix does not exist.

Matrix 3:

A=(301522418)A = \begin{pmatrix} 3 & 0 & 1 \\ 5 & 2 & 2 \\ 4 & 1 & 8 \end{pmatrix} We are asked to check the element A331A^{-1}_{33}, i.e., the element in row 3, column 3 of the inverse matrix. Using the cofactor method, we calculate the inverse, and find that: A331=213A^{-1}_{33} = \frac{2}{13} Thus, this matrix corresponds to B. A331=213A^{-1}_{33} = \frac{2}{13}.

Conclusion:

  • The first matrix corresponds to A. Det A = -36.
  • The second matrix corresponds to C. The inverse of the matrix does not exist.
  • The third matrix corresponds to B. A331=213A^{-1}_{33} = \frac{2}{13}.

Let me know if you need more details or explanations.

Here are some related questions to deepen understanding:

  1. How can you check if a matrix is invertible using its determinant?
  2. What are the properties of a matrix with a zero determinant?
  3. How do you calculate the cofactor of a matrix element?
  4. What methods can be used to find the inverse of a matrix?
  5. Can a matrix with negative determinant still be invertible?

Tip: For a 3x3 matrix, a nonzero determinant guarantees the existence of an inverse.

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Math Problem Analysis

Mathematical Concepts

Matrix Determinants
Matrix Inverses
Linear Algebra

Formulas

Determinant of a 3x3 matrix formula: Det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)
Inverse of a matrix: A^{-1} = 1/Det(A) * adj(A)

Theorems

A matrix has an inverse if and only if its determinant is non-zero.

Suitable Grade Level

Undergraduate level or advanced high school