Question Fill in the missing values in the matrix equation $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} = \begin{bmatrix} ? \\ ? \\ ? \end{bmatrix}$ .

Studdy Solution

STEP 1

Assumptions
1. We have a matrix multiplication problem.
2. The first matrix is a $3 \times 3$ identity matrix.
3. The second matrix is a $3 \times 1$ zero matrix.
4. We need to fill in the missing numbers in the resulting $3 \times 1$ matrix.

STEP 2

Recall the properties of the identity matrix. When any matrix is multiplied by an identity matrix of compatible dimensions, the result is the original matrix.
$I_n \cdot A = A$
where $I_n$ is the identity matrix of size $n \times n$ and $A$ is any matrix of size $n \times m$ .

STEP 3

Understand that a zero matrix, when multiplied by any other matrix, will result in a zero matrix of appropriate dimensions.
$A \cdot 0_{m \times p} = 0_{n \times p}$
where $A$ is any matrix of size $n \times m$ and $0_{m \times p}$ is a zero matrix of size $m \times p$ .

STEP 4

Apply the properties from STEP_2 and STEP_3 to the given problem. Since we are multiplying the identity matrix by a zero matrix, the result will be a zero matrix of the same dimensions as the second matrix.
$\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\left[\begin{array}{l} 0 \\ 0 \\ 0 \end{array}\right] = \left[\begin{array}{l} 0 \\ 0 \\ 0 \end{array}\right]$

STEP 5

Fill in the missing numbers in the resulting matrix with zeros.
$\left[\begin{array}{l} \square \\ \square \\ \square \end{array}\right] = \left[\begin{array}{l} 0 \\ 0 \\ 0 \end{array}\right]$
The missing numbers are all zeros.

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