Question

Let \(P=\left[a_{i j}\right]\) be a \(3 \times 3\) matrix and let \(Q=\left[b_{i j}\right]\), where \(b_{i j}=2^{i+j} a_{i j}\) for \(1 \leq i, j \leq 3\). If the determinant of \(P\) is 2 , then the determinant of the matrix \(Q\) is (A) \(2^{10}\) (B) \(2^{11}\) (C) \(2^{12}\) (D) \(2^{13}\)

Short answer

The determinant of the matrix Q is \(2^{10}\).

Step-by-step solution

Understand the Relationship Between Determinants of P and Q

The matrix Q is obtained by multiplying each element of matrix P by a factor of \(2^{i+j}\). The determinant of a matrix scales by the same factor if each element of a row (or column) is multiplied by that factor. Since each element of P is multiplied by a different power of 2 depending on its row and column indices, we can factor out these powers of 2 from each row (or column) when computing the determinant of Q.

Factor Out Powers of 2 From Each Row of Q

For each row in the matrix Q, we can factor out a power of 2. For the first row, factor out \(2^{1+1}\); for the second row, factor out \(2^{2+1}\); for the third row, factor out \(2^{3+1}\). In general, we factor out \(2^2\) from the first row, \(2^3\) from the second row, and \(2^4\) from the third row.

Calculate the Total Factor Taken Out of the Determinant

The total factor taken out from the determinant of Q by factoring out from each row is \(2^2\) for the first row, \(2^3\) for the second row, and \(2^4\) for the third row. Multiply these factors together to find the total factor that scales the determinant of P to give the determinant of Q: \(2^2 \times 2^3 \times 2^4 = 2^{2+3+4} = 2^9\).

Apply the Scaling Factor to the Determinant of P

Since the determinant of P is 2, and the determinant scales by the factor determined in the previous step, the determinant of Q is \(2 \times 2^9 = 2^{1+9} = 2^{10}\).

Key concepts

Determinant of a Matrix

The determinant of a matrix is a special number that can be calculated from its elements and provides important properties about the matrix, such as whether it's invertible or not. In essence, if you think of a matrix as a transformation applied to space, the determinant gives a sense of how much that transformation changes the volume in space.

For a simple 2x2 matrix, the determinant can be calculated by subtracting the product of its diagonals. However, for a 3x3 matrix like matrix P from our exercise, the determinant involves more steps, often employing a method called 'expansion by minors' or 'Laplace's formula'. This involves taking each element of a row (or column), and multiplying it by the determinant of the smaller matrix that remains when you erase the row and column of that element. Add and subtract these results according to a checkerboard pattern of signs to find the overall determinant.

It's crucial to note that the determinant of a matrix is a single value, despite the matrix having several elements. Additionally, only square matrices (matrices with the same number of rows and columns) have determinants. If a determinant of a matrix is zero, the matrix is considered singular, meaning it does not have an inverse.

Matrix Operations

Matrix operations encompass several operations that can be performed on matrices, such as addition, subtraction, scalar multiplication, and matrix multiplication. In our example, we're particularly interested in scalar multiplication. Scalar multiplication involves multiplying every element of a matrix by the same scalar value.

For instance, if we multiply matrix P by a scalar 2, every element in P is doubled. When the scalar is a power of 2, this is akin to a bit shift operation in binary terms. It's this principle that is used in the solution to the exercise, where each element is multiplied not just by a simple scalar, but by a scalar that is related to the position of the element within the matrix, represented as powers of 2.

Understanding scalar multiplication is fundamental in many areas of mathematics, including solving systems of linear equations, finding eigenvalues, and performing geometric transformations. Its implementation in algorithms can also be highly significant in computational efficiency.

Properties of Determinants

Determinants have various properties that make calculations more manageable and that can also offer insights into the characteristics of matrices. One such property states that the determinant of a matrix remains unchanged if a multiple of one row is added to another row. However, if a row is multiplied by a scalar, the determinant is multiplied by that scalar as well.

In relation to the exercise, when we change each element of matrix P by multiplying by a power of 2, we're utilizing a fundamental property of determinants. Specifically, the property that if each element of a row is multiplied by a factor 'k', the determinant of the entire matrix is multiplied by 'k'. Since we're dealing with all rows, the effect compounds.

Other properties include the fact that if two rows or columns of a matrix are identical, its determinant is zero. Also, if the matrix is invertible, its determinant is not zero. These properties often allow for shortcuts in computations and can make the difference between an efficient and an inefficient algorithm in computer implementations.