Question

Show that: a. det \(\left[\begin{array}{ccc}p+x & q+y & r+z \\ a+x & b+y & c+z \\ a+p & b+q & c+r\end{array}\right]=2\) det \(\left[\begin{array}{lll}a & b & c \\ p & q & r \\ x & y & z\end{array}\right]\) b. det \(\left[\begin{array}{ccc}2 a+p & 2 b+q & 2 c+r \\ 2 p+x & 2 q+y & 2 r+z \\ 2 x+a & 2 y+b & 2 z+c\end{array}\right]=9\) det \(\left[\begin{array}{lll}a & b & c \\ p & q & r \\ x & y & z\end{array}\right]\)

Short answer

a. det = 2 det b. det = 9 det

Step-by-step solution

Define Matrices Involved

Identify the matrices involved in both parts. For part (a), consider two matrices: \( A_1 = \begin{pmatrix} p+x & q+y & r+z \ a+x & b+y & c+z \ a+p & b+q & c+r \end{pmatrix} \) and \( A_2 = \begin{pmatrix} a & b & c \ p & q & r \ x & y & z \end{pmatrix} \). For part (b), consider matrices: \( B_1 = \begin{pmatrix} 2a+p & 2b+q & 2c+r \ 2p+x & 2q+y & 2r+z \ 2x+a & 2y+b & 2z+c \end{pmatrix} \) and the same \( A_2 \) as before.

Express Determinants

Rewrite the determinant expressions to identify any known properties. The determinant of matrix \( A_1 \) can be expanded utilizing the linearity of determinant along rows or columns, compare and relate it to the determinant of \( A_2 \). Similarly, express the determinant of \( B_1 \) in terms of combinations involving determinants, as multiple rows have linear combinations with parts from matrices.

Apply Determinant Properties for Part (a)

Utilize the property that indicates if two rows in a matrix change, the determinant relation to another matrix. For matrix \( A_1 \), split rows: \( \begin{pmatrix} p+x & q+y & r+z \ a+x & b+y & c+z \end{pmatrix} = \begin{pmatrix} p & q & r \ a & b & c \end{pmatrix} + \begin{pmatrix} x & y & z \ x & y & z \end{pmatrix} \) and adjust the third row accordingly. This results in doubling term combinations, hence \( 2 \cdot \text{det}(A_2)\).

Conclude Part (a)

Realize that by the multilinearity property of determinants and property due to row operations, \( \text{det}(A_1) = 2 \cdot \text{det}(A_2) \) holds due to the linear operations done on \( A_2 \).

Apply Properties for Part (b)

For matrix \( B_1 \), apply the similar property of linearity and constant multiplication: factor out constants from the matrix \( B_1 \) given each element in the original matrix is multiplied by 3. This results in a factor of \( 3^2 = 9 \) adjusting \( \text{det}(A_2) \), since the determinant is a multi-linear form.

Conclude Part (b)

The determinant of \( B_1 \) is calculated as \( 9 \cdot \text{det}(A_2) \) due to the structural alteration implying scaling by 3 of each element in the entire original matrix implies matrix product by 9, hence fulfilling \( \text{det}(B_1) = 9 \times \text{det}(A_2) \).

Key concepts

Matrix Algebra

Matrix algebra is a fundamental mathematical tool used to handle arrays of numbers organized in rows and columns. It simplifies complex systems, making them easier to analyze and solve. In the given exercise, matrices are employed to illustrate determinants, which help us understand properties such as invertibility and volume scaling in transformations. When dealing with matrices, basic operations include addition, multiplication, and scalar multiplication. These operations allow manipulation of matrices in various ways. For instance, if we have matrices \(A\) and \(B\), their sum \(C = A + B\) will have the same dimensions, while multiplication involves the dot product of rows of \(A\) with columns of \(B\). When these operations are completed accurately, they yield results fundamental to solving the problems involving matrices and their determinants.

Linearity of Determinants

The linearity of determinants is a key property that states how the determinant of matrices behaves under certain linear operations on their rows or columns. Let's break it down:

• It suggests the determinant of a matrix is linear concerning any row or column, meaning when you add or scale elements of a row, the determinant reflects these operations proportionally.
• For example, if \(A\) is a matrix and you modify one row by adding to it a scalar multiple of another row, the determinant will adjust in a predictable manner, yet remain invariant with operations like swapping rows.

In our exercise, this property allows us to analyze how determinants of matrices \(A_1\) and \(B_1\) are affected when their rows are linear combinations of rows from another matrix. By understanding such linear transformations, we rely on linearity to conclude the relation \( ext{det}(A_1) = 2 imes ext{det}(A_2) \).

Multilinearity Property

The multilinearity property of determinants extends linearity to multiple rows or columns simultaneously. This means the determinant behaves linearly with respect to each row and column independently, while all other rows and columns are held constant. This property can be very useful in:

• Understanding how mixing or splitting components in the rows of matrices influence the entire determinant.
• Enhancing the computation of determinants when matrices are constructed by inserting linear combinations of rows from various matrices.

Using it, we can comprehend complex matrices, like in the exercise where \(A_1\) and its split rows are linked to \(A_2\), further verifying \( ext{det}(A_1) \) as a distinct combination of \(A_2\)'s determinant.

Scaling Effect on Determinants

The scaling effect on determinants refers to how the determinant changes when a matrix is multiplied by a scalar. This effect is crucial:

• If all elements in a row (or column) of an \(n \times n\) matrix are multiplied by a scalar \(c\), the determinant of the matrix gets scaled by \(c\).
• For a full \(n \times n\) matrix where each element is multiplied by \(c\), the determinant is multiplied by \(c^n\).

In our exercise for matrix \(B_1\), each element was effectively tripled, reflecting as \(3^2\) in a 3x3 matrix determinant, exhibiting why \( ext{det}(B_1) = 9 \times ext{det}(A_2) \). Understanding this scaling helps us anticipate changes in the size and properties of the transformations represented by the matrix.