Let $\mathbf{A}=\left[a_{ij}\right]=\left[\begin{array}{cc} \log_5 128 & \log_4 5 \\ \log_5 8 & \log_4 25 \end{array}\right]$. If $\mathbf{A}_{ij}$ is the cofactor of $\mathbf{a}_{ij}$, $\mathbf{C}_{ij}=\sum_{k=1}^2 \mathbf{a}_{ik} \mathbf{A}_{jk}, 1 \leq i, j \leq 2$, and $\mathbf{C}=\left[\mathbf{C}_{ij}\right]$, then $8|\mathbf{C}|$ is equal to:

Step 1: Simplify the elements of matrix A

First, we simplify the elements of matrix $\mathbf{A}$ using logarithm properties. We express the numbers as powers of their bases and apply the change of base rule for logarithms, $\log_{b^m} a^n = \frac{n}{m} \log_b a$. Specifically, $\log_5 128 = \log_5 2^7 = 7\log_5 2$, $\log_4 5 = \log_{2^2} 5 = \frac{1}{2}\log_2 5$, $\log_5 8 = \log_5 2^3 = 3\log_5 2$, and $\log_4 25 = \log_{2^2} 5^2 = \frac{2}{2}\log_2 5 = \log_2 5$.

Step 2: Calculate the determinant of A

Next, we calculate the determinant of matrix $\mathbf{A}$. For a $2 \times 2$ matrix $\left[\begin{array}{cc} a & b \\ c & d \end{array}\right]$, the determinant is $ad-bc$. We use the property $\log_b a \cdot \log_a b = 1$. This simplifies the determinant calculation significantly.

Step 3: Relate $C_ij$ to the adjoint matrix

The definition of $\mathbf{C}_{ij}$ is given as $\sum_{k=1}^2 \mathbf{a}_{ik} \mathbf{A}_{jk}$. This sum represents the element in the $i$ -th row and $j$ -th column of the product of matrix $\mathbf{A}$ and the transpose of its adjoint matrix, $\text{adj}(\mathbf{A})^T$. However, a more direct interpretation is that if $i=j$, this sum is the determinant of $\mathbf{A}$. If $i \neq j$, this sum is 0.

Step 4: Determine the elements of matrix C

Based on the property of determinants, the sum $\sum_{k=1}^2 \mathbf{a}_{ik} \mathbf{A}_{jk}$ is equal to $|\mathbf{A}|$ if $i=j$ (diagonal elements) and $0$ if $i \neq j$ (off-diagonal elements). This means matrix $\mathbf{C}$ is a scalar multiple of the identity matrix, where the scalar is $|\mathbf{A}|$.

Step 5: Calculate the determinant of C

The determinant of a scalar multiple of an identity matrix is the scalar raised to the power of the matrix dimension. For a $2 \times 2$ matrix, $|k\mathbf{I}| = k^2$. Therefore, $|\mathbf{C}| = |\mathbf{A}|^2$. We substitute the value of $|\mathbf{A}|$ we calculated earlier.

Step 6: Calculate 8|C|

Finally, we multiply the determinant of $\mathbf{C}$ by 8 to get the required value.

More from this chapter

• Evaluate the determinant 1 & 0 & 4 3 & 5 & -1 0 & 1 & 2 (i) with the help of second row. (ii) with the help of third column.
• If A = [[1, 2], [3, 4]] find det(A) and adj(A).
• The value of 1 & 2 & 3 4 & 5 & 6 7 & 8 & 9 is:
• Using Cramer's rule, show that the following system of linear equations is consistent and hence solve it: x + y + z = 1, 2x + 3y + 2z = 2, x + y + 2z = 4.
• Let a R and A be a matrix of order 3 3 such that det(A)=-4 and A+I= 1 & a & 1 2 & 1 & 0 a & 1 & 2, where I is the 3 3 identity. If det ((a+1)adj((a-1)A))=2^m3^n
• Let A be a 3 × 3 matrix such that |adj (adj(adj A))| = 81. If S = n Z: |adj(adj A)|^(((n-1)²)/(2)) = |A|³n² - 5n - 4, then Σ_n S | A^(n² + n) | is equal to:
• Find the equation of the line joining A(1,3) and B(0,0) using determinants and find k if D(k, 0) is a point such that area of triangle ABD is 3 sq units.
• If the system of linear equations 3x + y + β z = 3, 2x + α y - z = -3, x + 2y + z = 4 has infinitely many solutions, then the value of 22β - 9α is:

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