The diagonal should be 31.2* the square root of 2 it’s the ratio of the sides of the 45 45 90 triangle formed when you divide a square in diagonal halves.
For the matrix
![\begin{bmatrix}{a_{11}} & {a_{12}} & {a_{13}} \\ {a_{21}} & {a_{22}} & {a_{23}} \\ {a_{31}} & {a_{32}} & {a_{33}}\end{bmatrix}](https://tex.z-dn.net/?f=%5Cbegin%7Bbmatrix%7D%7Ba_%7B11%7D%7D%20%26%20%7Ba_%7B12%7D%7D%20%26%20%7Ba_%7B13%7D%7D%20%5C%5C%20%7Ba_%7B21%7D%7D%20%26%20%7Ba_%7B22%7D%7D%20%26%20%7Ba_%7B23%7D%7D%20%5C%5C%20%7Ba_%7B31%7D%7D%20%26%20%7Ba_%7B32%7D%7D%20%26%20%7Ba_%7B33%7D%7D%5Cend%7Bbmatrix%7D)
the determinant using the method of expansion by minors, expanding on the third row is:
![\det \begin{bmatrix}{a_{11}} & {a_{12}} & {a_{13}} \\ {a_{21}} & {a_{22}} & {a_{23}} \\ {a_{31}} & {a_{32}} & {a_{33}}\end{bmatrix}=a_{31}\det \begin{bmatrix}{a_{12}} & {a_{13}} & {} \\ {a_{22}} & {a_{23}} & {} \\ {} & {} & {}\end{bmatrix}-a_{32}\det \begin{bmatrix}{a_{11}} & {a_{13}} & {} \\ {a_{21}} & {a_{23}} & {} \\ {} & {} & {}\end{bmatrix}+a_{33}\det \begin{bmatrix}{a_{12}} & {a_{12}} & {} \\ {a_{22}} & {a_{22}} & {} \\ {} & {} & {}\end{bmatrix}](https://tex.z-dn.net/?f=%5Cdet%20%5Cbegin%7Bbmatrix%7D%7Ba_%7B11%7D%7D%20%26%20%7Ba_%7B12%7D%7D%20%26%20%7Ba_%7B13%7D%7D%20%5C%5C%20%7Ba_%7B21%7D%7D%20%26%20%7Ba_%7B22%7D%7D%20%26%20%7Ba_%7B23%7D%7D%20%5C%5C%20%7Ba_%7B31%7D%7D%20%26%20%7Ba_%7B32%7D%7D%20%26%20%7Ba_%7B33%7D%7D%5Cend%7Bbmatrix%7D%3Da_%7B31%7D%5Cdet%20%5Cbegin%7Bbmatrix%7D%7Ba_%7B12%7D%7D%20%26%20%7Ba_%7B13%7D%7D%20%26%20%7B%7D%20%5C%5C%20%7Ba_%7B22%7D%7D%20%26%20%7Ba_%7B23%7D%7D%20%26%20%7B%7D%20%5C%5C%20%7B%7D%20%26%20%7B%7D%20%26%20%7B%7D%5Cend%7Bbmatrix%7D-a_%7B32%7D%5Cdet%20%5Cbegin%7Bbmatrix%7D%7Ba_%7B11%7D%7D%20%26%20%7Ba_%7B13%7D%7D%20%26%20%7B%7D%20%5C%5C%20%7Ba_%7B21%7D%7D%20%26%20%7Ba_%7B23%7D%7D%20%26%20%7B%7D%20%5C%5C%20%7B%7D%20%26%20%7B%7D%20%26%20%7B%7D%5Cend%7Bbmatrix%7D%2Ba_%7B33%7D%5Cdet%20%5Cbegin%7Bbmatrix%7D%7Ba_%7B12%7D%7D%20%26%20%7Ba_%7B12%7D%7D%20%26%20%7B%7D%20%5C%5C%20%7Ba_%7B22%7D%7D%20%26%20%7Ba_%7B22%7D%7D%20%26%20%7B%7D%20%5C%5C%20%7B%7D%20%26%20%7B%7D%20%26%20%7B%7D%5Cend%7Bbmatrix%7D)
Answer:
First, we compute the determinants of the minors:
![\begin{gathered} \det \begin{bmatrix}{0} & {4} & {} \\ {-1} & {3} & {} \\ {} & {} & {}\end{bmatrix}=0+4=4, \\ \det \begin{bmatrix}{1} & {4} & {} \\ {1} & {3} & {} \\ {} & {} & {}\end{bmatrix}=3-4=-1, \\ \det \begin{bmatrix}{1} & {0} & {} \\ {1} & {-1} & {} \\ {} & {} & {}\end{bmatrix}=-1-0=-1. \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20%5Cdet%20%5Cbegin%7Bbmatrix%7D%7B0%7D%20%26%20%7B4%7D%20%26%20%7B%7D%20%5C%5C%20%7B-1%7D%20%26%20%7B3%7D%20%26%20%7B%7D%20%5C%5C%20%7B%7D%20%26%20%7B%7D%20%26%20%7B%7D%5Cend%7Bbmatrix%7D%3D0%2B4%3D4%2C%20%5C%5C%20%5Cdet%20%5Cbegin%7Bbmatrix%7D%7B1%7D%20%26%20%7B4%7D%20%26%20%7B%7D%20%5C%5C%20%7B1%7D%20%26%20%7B3%7D%20%26%20%7B%7D%20%5C%5C%20%7B%7D%20%26%20%7B%7D%20%26%20%7B%7D%5Cend%7Bbmatrix%7D%3D3-4%3D-1%2C%20%5C%5C%20%5Cdet%20%5Cbegin%7Bbmatrix%7D%7B1%7D%20%26%20%7B0%7D%20%26%20%7B%7D%20%5C%5C%20%7B1%7D%20%26%20%7B-1%7D%20%26%20%7B%7D%20%5C%5C%20%7B%7D%20%26%20%7B%7D%20%26%20%7B%7D%5Cend%7Bbmatrix%7D%3D-1-0%3D-1.%20%5Cend%7Bgathered%7D)
Therefore:
The thickness of each book was A) 2.25 inches
Answer:
Step-by-step explanation:
x=0.6121212... (1)
10x=6.121212... (2)
1000 x=612.121212... (3)
(3)-(2) gives
990 x=606
330x=202
165x=101
x=(101)/165