Show that matrix A is the inverse of matrix B. A=[ 3 8 2 1 1 1 5 6 12 ],

step 1 So we're given two matrices, A and B. And we want to show that A is the inverse of V. That is, w

Show that matrix A is the inverse of matrix B. A=[ 3 8...

Show that matrix $A$ is the inverse of matrix $B$.
$$A=\left[\begin{array}{lll}{3} & {8} & {2} \\ {1} & {1} & {1} \\ {5} & {6} & {12}\end{array}\right], B=\frac{1}{36}\left[\begin{array}{ccc}{-6} & {84} & {-6} \\ {7} & {-26} & {1} \\ {-1} & {-22} & {5}\end{array}\right]$$

Key Concepts

Identity Matrix

The identity matrix is a special kind of square matrix that has ones on the diagonal and zeros elsewhere. It acts as the multiplicative identity in matrix algebra, meaning any matrix multiplied by the identity matrix remains unchanged. In verifying inverse relationships, the product of a matrix and its inverse should equal the identity matrix, confirming the correctness of the inversion.

Matrix Multiplication

Matrix multiplication is a binary operation that produces a new matrix from two matrices by taking the dot product of rows and columns. In the context of inverses, multiplying a matrix by its inverse is supposed to give the identity matrix, which validates that one matrix is indeed the inverse of the other. This process involves systematically summing the products of corresponding elements in rows and columns.

Matrix Inverse

A matrix inverse is a matrix that, when multiplied by the original matrix, yields the identity matrix. It is a fundamental concept in linear algebra that provides a way of solving systems of linear equations and understanding the behavior of linear transformations. The existence of an inverse is tied to the determinant of the matrix being nonzero, and not all matrices have an inverse.

Transcript

00:01 So we're given two matrices, a and b.

00:03 And we want to show that a is the inverse of v.

00:06 That is, we want to show that a is b raised to the negative 1.

00:11 So if a is the inverse matrix of b, then we know that a times b must equal the identity matrix for a3 by 3, which is just represented by a diagonal of 1, zeros on this side, and zeros on this 1.

00:30 So this is the identity matrix for a 3x3.

00:34 So if we multiply a times b and we obtain this matrix right here, we have proven that a is the inverse matrix of b.

00:44 So to perform these multiplications, we're going to take this first row of a and multiply it times the first column of b to obtain the first term of the first row, sorry, which is this one right here.

01:04 Then we're going to multiply the first row over here times the second column to obtain the second term of the first row on the new matrix.

01:14 Finally, we're going to repeat the process where our third column to find the third element in our first row.

01:22 And we're going to repeat this process with the remaining rows in matrix a.

01:29 So, this multiplication will look like 3 times negative 6 is just negative 18.

01:37 Plus, so we're going to add all of these terms, 8 times 7, which is just 56, and 2 times negative 1, which is just negative 2.

01:47 I'm going to draw a line over here, just so we can have enough space to determine which terms belong to which row and column.

01:57 Now we're going to multiply 3 times 84, which is just 252.

02:06 Now we're going to multiply 8 times negative 26, which is just negative 208.

02:14 And finally, 2 times negative 22, which is just negative 44.

02:21 Now if we do that with the third column, we're going to obtain 3 times negative 6 is just negative 18.

02:29 8 times 1 is just 8.

02:31 And then 2 times 5 is just 10, sorry.

02:40 Now for the second row, sorry, we have 1...

Please give Ace some feedback